For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

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## Transcription

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### Introduction to Logarithms

- A
*logarithm*is a way to reverse the process of exponentiation. It allows us to mathematically ask the question, "Given some base and some value, what exponent would we have to use on the base to create that value?" - The logarithm base a of x (log
_{a}x), where a > 0 and a ≠ 1, is defined to be the number y such that a^{y}= x.log _{a}x = y ⇔ a^{y}= x. - The idea of a logarithm can be really confusing the first few times you work with it, so make sure to watch the video to clarify how logarithms are used and what they mean.
- The two most common logarithmic bases to come up are the numbers e and 10. As such, they have special notation because we have to write them so often.
- The base of e is expressed as ln. It is called the
*natural logarithm*. [Remember, e is called the*natural base*.]lnx ⇔ log _{e}x - The base of 10 is expressed with just log: if no base is given, it is assumed to be base 10. It is called the
*common logarithm*.logx ⇔ log _{10}x

- The base of e is expressed as ln. It is called the
- Just like exponentiation, we can find the value (or a very good approximation) of a logarithm by using a calculator. Any scientific or graphing calculator will have ln and log buttons to take logarithms base e and 10, respectively. However, many calculators will not have a way to take logarithms of arbitrary bases. There is a way around this called
*change of base*, and we'll explore it in the next lesson,*Properties of Logarithms*. - When we graph logarithms, we see they grow very slowly. (This is because they are the inverse of exponentiation.) The graph of a logarithm also approaches the y-axis asymptotically. It gets very close, but it doesn't touch it.
- Logarithms are the inverse of exponentiation, and vice-versa. The exponential function of base a is the inverse of the logarithmic function of base a:
f(x) = log _{a}x f^{−1}(x) = a^{x} - Because a logarithm is the inverse of exponentiation, we cannot take the logarithm of some numbers. Exponentiation (a
^{x}) only has a range of (0,∞), so the corresponding logarithm can only reverse those same output values. In other words,Domain of log _{a}x: (0, ∞) . - We can also see the above domain must be true because it would make no sense otherwise. Consider that log
_{2}0 = b means that 2^{b}= 0. But no such number b exists that could do this! The same goes for negative numbers, so the domain of any logarithm must be (0, ∞).__Note:__This means we have now introduced a new type of thing to watch out for when we're looking for domains. Before, we only had to worry about dividing by 0 and having negatives under a square root. (Depending on what you've done previously, you might also have needed to be careful about certain things with trigonometric functions.) Now we also have to watch out for taking a logarithm of 0 or a negative number. One more thing to pay attention to when you're finding the domain of a function.

### Introduction to Logarithms

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Introduction 0:04
- Definition of a Logarithm, Base 2 0:51
- Log 2 Defined
- Examples
- Definition of a Logarithm, General 3:23
- Examples of Logarithms 5:15
- Problems with Unusual Bases
- Shorthand Notation: ln and log 9:44
- base e as ln
- base 10 as log
- Calculating Logarithms 11:01
- using a calculator
- issues with other bases
- Graphs of Logarithms 13:21
- Three Examples
- Slow Growth
- Logarithms as Inverse of Exponentiation 16:02
- Using Base 2
- General Case
- Looking More Closely at Logarithm Graphs
- The Domain of Logarithms 20:41
- Thinking about Logs like Inverses
- The Alternate
- Example 1 25:59
- Example 2 30:03
- Example 3 32:49
- Example 4 37:34

### Math Analysis Online

### Transcription: Introduction to Logarithms

*Hi--welcome back to Educator.com.*0000

*Today, we are going to have an introduction to logarithms.*0002

*At this point, if we want to find the value of a number raised to an exponent, it is easy.*0005

*We use our exponentiation rules and evaluate; if it is something simple, like 2 ^{3},*0008

*then we know that is 2 times 2 times 2, 2 times itself 3 times, which we figure out is 8.*0012

*And we also figured out how to do numbers that weren't just simple integer exponents.*0018

*But we have all of these nice rules from our previous few lessons.*0022

*But what if the question was inverted, and what if we knew the base and the end result, but we don't know the exponent that we need to get there?*0025

*If we knew that we had 2 as the base, and we wanted to have an end result of 8,*0033

*but we had no idea what exponent we had to use to get to 8, how could we figure out that exponent?*0038

*This is the question that leads us to explore the idea of the logarithm, which we will be looking at over the next few lessons.*0045

*We define the logarithm as a way to talk about this unknown exponent.*0052

*The logarithm base 2 of x, denoted log _{2}(x), with this little 2 as a subscript right here,*0055

*is defined to be the number y such that 2 ^{y} = x.*0065

*So, when we see log _{2}(x) = y, then we know that that would be 2^{y} = x.*0071

*In other words, we are looking for the number that...when the logarithm goes onto some number,*0079

*we are trying to figure out what value, if it raised to this space here, would become the number that we took the logarithm of.*0085

*So, for the example of log _{2}(8) = 3, the reason why that is the case is because*0094

*we are looking for what number here we have to raise to to get 8.*0100

*So, the answer there is 3; if we raise 2 ^{3}, we get 8, so we know that log_{2}(8) = 3.*0105

*It can be a little bit confusing to remember how this works at first--what the notation means.*0115

*So, when you see log _{2}(x) = y, you can think of this as...if the 2 were under the y,*0120

*if we had 2 ^{y}, it would make the x that we are taking the logarithm of.*0129

*So, we take the logarithm of some number in regards to some base.*0135

*And that tells us what number we would raise the base to, to get the original number we are taking the logarithm of.*0139

*It will make more sense as we see more examples.*0146

*Here are some other examples: log _{2}(32) = 5, because if we raise 2 to the 5^{th}, we get 32.*0149

*2 times 2 times 2 times 2 times 2 is equal to 32; 4, 8, 16, 32.*0158

*log _{2}(1) = 0 because, if we raised 2 to the 0, from our rules about exponents, we know that that is just the same thing as 1.*0168

*log _{2}(1/4) = -2, because we know that, if we raise 2 to the -2, then that is the same thing as 1/2 raised to the 2,*0177

*where you flip to the reciprocal; and 1/2 squared would become 1/4, which is what we initially started with.*0186

*So, a logarithm is a way of taking a logarithm of a number, so that you figure out*0194

*what you would have to raise some base to, to get the thing you took the logarithm of.*0199

*We can expand this idea to something beyond just base 2, to a general idea.*0204

*The logarithm base a of x, log _{a}(x) (that little a, right down here, is a subscript), where a > 0,*0209

*and a is not equal to 1 (our base has to be greater than 0, and our base has to not be equal to 1),*0218

*is defined to be the number y such that a ^{y} = x.*0224

*If we take log _{a}(x), then we know that that gives us some y, and that a^{y} = x.*0228

*So, once again, it is the same idea, where, if we take this base, and we put it under the y, we would get a ^{y}.*0236

*And then, we would have the thing that we were originally taking the logarithm of, which is what we have there.*0245

*That is what is occurring right here.*0249

*The idea of the logarithm is that you take the log, and it tells you something that you can raise a number to, to be able to get this other value.*0251

*It is a little bit complex the first time you get it; but as you do it more and more, it will start to make more sense.*0260

*And we are going to see a whole bunch of examples to really get this cleared up.*0264

*Now, notice: we have these restrictions on what our base can be.*0267

*We know that the base has to be greater than 0, and the base is not equal to 1.*0271

*The base a of a logarithm has the same restrictions as the base of an exponential function.*0274

*This is because exponentiation and logarithms are inverse processes; they do the opposite thing.*0280

*And we will see more about how they are inverses in the future.*0288

*But they do reverse things, so they have to have the same restrictions, because they are basically working with the same idea of a base.*0291

*They are being seen through different lenses, and it will make more sense as we work on it more and more;*0299

*but we have to have the same restrictions on it--otherwise the idea of a logarithm will just not make sense, or not be very interesting.*0303

*So, we have these restrictions: that we have to have our base greater than 0, and we have to have our base not equal to 1.*0309

*Let's look at some examples to help clear this idea up.*0316

*log _{7}(49) = 2, because, if we move this base over,*0319

*then 7 ^{2} is just 7 times 7, which gets us 49, which is exactly what we started with.*0324

*So, this is the case, because 7 ^{2} = 49; or maybe let's write it in the way that we had it originally here: 49 =...*0332

*moving the base over, moving our base underneath the right side...we have 7 ^{2}, like this.*0343

*The same thing over here: we know that log _{10}(10000) = 4,*0350

*because, if we move our base under, we know that 10 ^{4} is equal to 10000.*0354

*The question is: if we want to know what number we have to raise 10 to, to get 10000--that is what log _{10}(10000) is effectively asking.*0363

*It is saying, "What number do we have to raise 10 to--raise our base to--to get 10000 as the end result?"*0379

*10 to the what equals 10000? The answer to that is 4.*0387

*So, we take 10 ^{4}; we get 10000; and sure enough, we see that that is 10 times 10 (100) times 10 (1000) times 10 (10000).*0391

*The same thing is going on over here: if we have log _{5}(1/125), then we move that under, and we get 5^{-3} = 1/125.*0400

*Let's check that out: 5 ^{-3} is the same thing as 1/5 to the positive 3; we flip to the reciprocal.*0413

*And then, 5 times 5 times 5 is 25...125; so we get 1/125; sure enough, it checks out.*0423

*Finally, if we take log _{4}(2), then we see that 4^{1/2} is equal to 2.*0433

*What is 1/2? 1/2 is an exponent that means square root; 4 ^{1/2} is the same thing as √4, which is 2; so once again, that checks out.*0441

*It is the question of what exponent I am looking for to be able to get this base to become the number that I am taking the logarithm of.*0451

*We can even do this with more unusual bases on our logarithms.*0459

*For example, if we have log _{1/2}(1/16), then we can see that that will become 4, because (1/2)^{4} is equal to 1/16,*0462

*because 2, 4, 8, 16...1/2, 1/4, 1/8, 1/16; so we see that that is the same thing.*0475

*If we take log _{π}(π), then we know that it has to come out as 1,*0486

*because π ^{1}--of course, that is no surprise that it is going to equal what it already started with.*0492

*So, if we are taking log _{π}(π), then the thing that π has to be raised to, to get π, is just 1.*0499

*So, we have 1 as the thing that comes out of that.*0505

*If we take log _{√2}(4√2), then we get that that has to be 5, because (√2)^{5}--sure enough, that is equal to 4√2.*0509

*We can check that out: √2 times √2 times √2 times √2 times √2.*0524

*Well, √2 times √2 becomes 2; √2 times √2 becomes 2; and we have this one √2 left.*0531

*So, 2 times 2 is 4, and we are left with 4√2; it checks out.*0537

*The final one: log _{e}(1) becomes 0, because if we move this over, e^{0}, just like anything raised to the 0, becomes 1.*0542

*So, it checks out; so we have some idea of how it works.*0553

*A logarithm is, "For this base, what number do I have to raise it to, to get the number that I am originally taking the logarithm of?"*0555

*When I take the log _{10}(10000), it is a question of what number I have to raise 10 to, to get 10000.*0562

*I have to raise 10 to 4 to get 10000.*0570

*When I take log _{7}(49), it is a question of what number I need to raise 7 to, to get 49; the answer to that is 2.*0574

*That is the idea of a logarithm.*0582

*The two most common logarithmic bases to come up are the numbers e (remember, e is the natural number;*0585

*we talked about it previously, when we talked about exponential functions--a very important idea) and 10.*0591

*As such, they have special notation, because we have to write them so often.*0597

*The base of e is expressed as ln; so when we want to talk about base of e, the shorthand for that is ln.*0601

*It is called the natural logarithm; remember, e is called the natural base.*0607

*So, when we are taking a log _{e}, we call it a natural logarithm, and we use ln, because originally,*0612

*the French were the ones who came up with this; so it was logarithme naturel (excuse my French--I am not very good at speaking French).*0617

*So, the natural log of x is equivalent to log _{e}(x); ln(x) is just a shorthand way of saying log with a base of the number e.*0623

*Base 10 is expressed with just log on its own; notice, it has no subscript--there is no little number down there.*0634

*If no base is given, it is assumed to be base 10; since base 10 comes up a lot, it is just an easy way to write it; this is normally what it means.*0641

*It is called the common logarithm, because it is a commonly-used logarithm.*0648

*So, if you see log(x), notice that it has no little subscript--no little number down there.*0652

*Then, we know that that is going to mean log _{10}(x).*0657

*Well, we can find the value of expressions like log _{2}(8); we know that that came out to be 3,*0662

*because the number that we raised 2 to, to get 8, is 3.*0667

*How do we figure out the value of more complicated expressions?*0670

*Like if we wanted to figure out the natural log of 12.19--and as we just saw, that would be the same thing as asking, "What is log _{e} or 12.19?"*0673

*Well, e is a very complicated number; it goes on forever--it is irrational.*0681

*12.19 is not a very friendly--looking decimal number; so how are these two things going to interact?*0685

*We can guess that it is probably not going to come out very cleanly, in a nice way.*0690

*Sure enough, it doesn't: it comes out to be approximately 2.500616; and precisely, it would keep going forever, as well.*0693

*So, for calculating logarithms, just like exponentiation, we can find the expressions, or a very good approximation, by using a calculator.*0701

*Any scientific or graphing calculator will have natural log and log _{10} buttons to take logarithms base e and 10, respectively.*0709

*However, many calculators will not have a way to take logarithms of arbitrary bases.*0718

*So, if we had log _{3}, most calculators won't have an easy way for us to just get what log_{3}(some number) is.*0723

*But there is a way around this, and it is called change of base.*0731

*So, if you do need to take the log base 3 of some number, check out the next lesson, Properties of Logarithms,*0734

*where we will explore how you can change from one base to another.*0740

*So, the way that you calculate complicated logarithms like this is: you generally just use a calculator.*0743

*The calculator has a way, a method, to be able to figure out what that comes out to be.*0748

*Now, just like with exponentiation that we talked about before, we should note that there are ways to calculate these values by hand.*0753

*We could do this by hand and figure it out; and you will learn about this in more advanced math classes.*0759

*But we won't learn about it in this course right here.*0764

*Doing this takes a lot of arithmetic, though; and so we designed calculators to speed up the process.*0766

*It is something that we could do; it is not like we are completely reliant on calculators for figuring this idea out.*0771

*Logarithms weren't something that we only got once we had calculators created.*0776

*We have been able to have this idea for a very long time--since the 1600s, in fact.*0780

*But being able to calculate what these numbers come out to be--that takes a long time; it is a slow process.*0785

*So, we have calculators to be able to figure this out for us very quickly and very easily.*0790

*So, it speeds things up, but it is not that we are dependent on calculators.*0794

*It is just that they are a useful tool that we can apply in this situation.*0797

*Graphs of logarithms: so now, since we can evaluate logarithms however we want,*0802

*because we have these nice calculators as tools, we can plot graphs of them.*0806

*So, let's look at some graphs: f(x) = log _{2}(x) (this is in red); g(x) = log_{5}(x)*0809

*(that one is in blue), and finally h(x) = log _{10}(x) (that one is in green).*0817

*Notice how short the y-axis is; it only goes from -3 up until positive 5.*0823

*But we go all the way out to positive 100 on the x-axis.*0830

*We can see that here--right here; it is hard to see--that is a 1 value on the x-axis.*0835

*And that is going to end up corresponding to 0, because log of anything--log of any base of 1--will come out to be 0,*0842

*because the number that you raise anything to, a ^{0}, = 1.*0855

*So, if we would take log base anything of 1, it is going to always come out to be 0,*0859

*because that is the number that we raise anything to, to get 1 in the first place.*0867

*So, that is why we see a common height of 0 there.*0871

*And notice how slowly they grow: at 16, log _{2} is only going to be at a meager 4.*0873

*But for log _{10}, when we look at log_{10}, it takes getting all the way up to 100 to even get to 2.*0881

*If we go out here to the 2, it takes all the way to 100 to be able to get that from log _{10}, because 10^{2} = 100.*0889

*We are seeing a similar thing for log _{5}: it has to get all the way up to 25 before it hits this height of 2, as well, because it is 5^{2}.*0899

*And we aren't even going to see it hit height 3, because it is not going to hit a height of 3 until it manages to get to 125*0909

*as an input value, because 5 ^{3} becomes 125.*0915

*So, notice how slowly these graphs grow.*0919

*These graphs grow really, really slowly, because for logs, it takes a really big number to be able to get even slight increases in our verticals.*0923

*And the farther out they go, the even slower they are going to grow.*0930

*Now, notice that they approach the y-axis asymptotically.*0933

*So, as they get smaller and smaller, they get really, really close to this y-axis right here.*0937

*They never touch or pass it, although that might be hard to see in this picture, since it looks like it is right on top of it.*0943

*But they get very close; they won't actually touch it, but they get very close to the y-axis.*0948

*We will talk about this behavior of how it gets really close to the y-axis,*0954

*and why it can't actually touch the y-axis or go past it, soon, when we talk about the domain of a logarithm.*0957

*The logarithm is the inverse process of exponentiation.*0963

*For example, let's consider base 2: if we have log _{2}(x) = y, then we can see its flip of 2^{y} = x.*0966

*We just change the x and the y location.*0973

*So, if we take log _{2}(8), that becomes 3, because remember: 2^{3} = 8.*0975

*So, when we take log _{2}(8), we get 3.*0982

*But then, if we take that 3, and we plug it into the other one--we take the 3, and we plug it in up here--*0985

*we look at 2 ^{3}--look: we are right back where we started.*0990

*We have the same thing on both sides.*0993

*We take this log, and we do something to it, and then we do the reverse process with the same base as the exponentiation.*0997

*We get back to the original input that we put in.*1003

*The same thing: we did it the other way, where we did exponentiation first.*1006

*If we take 2 ^{-2}, then that is going to flip to (1/2)^{2}; so we would get 1/4.*1009

*And then, if we take log _{2}(1/4), we are going to get -2.*1013

*So, exponentiation and logarithms are doing inverses: one goes one way, and one goes the other way.*1018

*Together, they cancel out; we will be discussing this idea a lot more in the coming lessons.*1023

*It is a very important thing; we will also be proving it in general.*1027

*We can see this as one in a general form for any logarithm.*1030

*The exponential function of base a is the inverse of the logarithmic function of base a.*1034

*It is critical, though, that they do have the same bases; our exponential function is base a, and our logarithmic function must be base a.*1039

*If they are not the same base, it won't work.*1046

*Let's see why this is the case: if we have f(x) = log _{a}(x), f^{-1}(x) = a^{x}.*1048

*Then, we can take f ^{-1}(f(x)) and see what happens.*1055

*Now, remember: we are talking about stuff from our lesson on inverse functions.*1061

*If you need more background on inverse functions, make sure you go and check out that lesson.*1065

*It will help you understand what is going on here.*1068

*So, f ^{-1}(f(x)) =...well, we could do this as...since this is a^{x}, then it is going to be...*1071

*well, we will apply f ^{-1} next; first, f(x) = log_{a}(x).*1077

*We have log _{a}(x); then we apply the f^{-1}, and we have a^{loga(x)}.*1082

*Now, what does that end up coming out to be?*1092

*Well, remember: log _{a}(x) = y is the same thing as saying a^{y} = x.*1094

*So, that is what log _{a}(x) is: it is this y, some y such that if we were to put it as an exponent on a, we would get x.*1104

*So, log _{a}(x) = y: we can just say, "Whatever the number log_{a}(x) is, let's call it y."*1115

*So, we can swap that out, and we can say, "a ^{y}," just because we are saying we will call log_{a}(x) y.*1120

*That is what we have over here; but remember, we defined this idea of what log _{a}(x) is based on a^{y} = x.*1126

*Well, we now have a ^{y} = x; so if a^{y} = x, then that equals x,*1133

*which means that f ^{-1}(f(x)) = x.*1139

*Whatever we put in as our input comes out as our output if we do these two functions, one on top of the other.*1144

*We have inverse functions, because one function cancels out the effects of the other function.*1149

*We will talk about this more in future lessons.*1154

*We can also see this in the graphs of exponential and logarithmic functions.*1157

*So, if we take two graphs of, say, 2 ^{x} (that one is in red) and log_{2}(x), we see them like this.*1161

*And then finally, we also have y = x in yellow here, coming through the middle.*1168

*Now, remember from our lesson about inverse functions: when we learned about inverse functions,*1174

*we know that if two functions are inverses, they mirror over the line y = x.*1180

*They are swapping x and y coordinates; this shows us that they have to be inverses.*1188

*For example, if we look at what log _{2} at 2 is, it comes out to be a height of 1; here is a height of 1.*1193

*And then, if we look at what our 2 ^{x} at 1 is, at 1 it is a height of 2.*1201

*So, for this one, we have (I'll color-code it back to what it had been) (1,2).*1212

*But for the blue one, we have (2,1).*1223

*They flip x and y locations, and that is going to be true wherever we go on this, because we see that they do this thing with y = x,*1226

*where they mirror across it; their x and y locations swap, showing us that they are inverses.*1236

*Notice all the graphs that we have seen of logarithms: they never pass, or even touch, the y-axis.*1243

*They never pass the y-axis; they never even manage to touch the y-axis.*1248

*This is because the domain of a logarithm is 0 to infinity.*1252

*And notice: there is a parenthesis on that 0; so it says it is not inclusive--*1257

*so, not including 0, everywhere up from 0 (but not including 0), all the way up to positive infinity.*1260

*We can see this for a couple of reasons.*1266

*First, since exponentiation and logarithms are inverses, that means that the range of an exponential function is the domain of a logarithm.*1268

*The range of f(x) = a ^{x} is going to be 0 to infinity.*1275

*a ^{x}...if we put in any base a that is greater than 0 and not 1, it is going to go anywhere from 0 up until infinity.*1280

*If we look at 2 ^{x}, by varying what we plug in for x, we are going to be able to get anything between 0 and positive infinity.*1288

*Now, let's talk briefly about this idea: if we had a pool of numbers that we called a, the set of things that we are allowed to use,*1297

*and then we had another pool of numbers that was b, the set of things that it is possible to get to through some function f...*1306

*we have some function f that takes numbers from a, and it goes to b; then we call the numbers over here domain.*1315

*We talked about this when we first talked about the ideas of functions.*1322

*So, here is the domain of f; and over here is the range of f.*1326

*The domain of f is everything that f is able to take in; the range of f is everything that f is able to put out.*1331

*So, for the example a ^{x} or the example 2^{x}, as a specific example,*1338

*the domain is anything; it can take in any number at all--negative infinity to positive infinity--any real number whatsoever.*1343

*But it is only going to be able to give out numbers from 0 to infinity.*1352

*So, in this case, we see that it is going to have its range as 0 to infinity.*1355

*Now, notice: if we do the reverse of this, if we want to see the reverse of this, a function that does the opposite of what f does,*1359

*f ^{-1}, then it is going to have to go, not from a, but from b.*1368

*So, its domain, the domain of f ^{-1}, is going to be going in the other direction.*1373

*Since it is taking what f did and reversing it, it has to be able to take the things that f does as outputs.*1380

*Whatever f makes as outputs--whatever f puts out--is what f ^{-1} will take in.*1385

*So, the domain of f ^{-1} is the range of f, which means that the range of f^{-1} is also the domain of our original function, f.*1391

*f goes from a to b; f ^{-1} goes from b to a.*1402

*Now, we saw that, for any exponential function, its range is 0 to infinity.*1407

*So, that means that the domain of f ^{-1}, the domain of a logarithmic function,*1412

*since it is the inverse of exponentiation, must also be from 0 to infinity.*1419

*So, that is why we have this domain here; the domain of any log has to be from 0 to infinity,*1425

*because the range of any exponential function is from 0 to infinity.*1430

*So, they are going to be done as opposites; the range of an exponential function is the domain of a logarithmic function.*1434

*So, that is a fancy way to be able to understand why this has to be the case,*1441

*because we can say what we learned about inverse functions applies here, because we have an inverse function.*1446

*But alternatively, we can just see that it would not make sense--it just is nonsense if we look at it otherwise.*1450

*Consider if we tried to take log _{2}(0); then we know that that has to be equal to some number b for it to be a possible thing.*1456

*Then that means that 2 ^{b} has to somehow be equal to 0.*1463

*But that doesn't make any sense: no such number b exists.*1470

*No possible number could exist that would be able to take 2 and turn it into 0.*1474

*2 ^{b} can't ever become 0; if we plug in any number, we can make very small numbers; but we can't actually get all the way to 0.*1480

*We can't touch 0; the same is going to go for negative numbers.*1486

*If we wanted to say 2 ^{b} = -4, there is no number that does that.*1489

*We can't raise 2 to some number and make it negative--it started out positive, so we can't possibly make it negative.*1495

*So, this is impossible; this is impossible; this is impossible; so it means that log _{2}(0) is an impossible idea.*1501

*We can't take the logarithm of 0 or anything that is going to be negative,*1509

*because it just won't be possible for it to work over here, where we are trying to figure out what would be the exponential version of it.*1513

*So, since it just doesn't make sense to take the logarithm of a number that is 0,*1519

*or to take the logarithm of a number that is negative, it must be that the domain is always positive.*1523

*We have to go from 0, but not including 0, all the way up to positive infinity.*1528

*We can take in any of those things, but we can't take in 0; we can't take in negative numbers.*1533

*That explains why our domain has to be this.*1537

*We can think about it that way, or we can think about it as this flipped idea of the fact that exponentiation and logarithms are inverses.*1540

*So, we can have this more complex idea of the domain and range of what those things have to be.*1546

*But we can also just go to the fact that it does not make sense--it would be nonsense; that is a reasonable idea, too.*1552

*All right, we are ready for some examples.*1559

*Let's evaluate these numbers without a calculator.*1561

*If we are looking at log _{6}(216), then that is going to be equal to some number,*1564

*such that, when we raise 6 to that number, we get 216.*1568

*216 = 6 ^{?}: we want to figure out what this is, so let's see.*1572

*What are some numbers that we could get out of this?*1577

*6 ^{1}--that is just 6; 6^{2}--well, that would be 36; 6^{3} is 180 + 36 = 216.*1579

*That is what we are looking for; so it must be the case that it is 3: log _{6}(216) = 3; that is our answer.*1588

*If we have log(1/10000), the first thing we want to do is remember: if we have just log, then that is a way of saying it is log base 10.*1600

*So, log _{10} (1/10000)--once again, we are asking what that is going to be.*1609

*Well, that is going to be the number such that 10 to whatever that number is is going to be equal to 1/10000.*1613

*So, let's look at possible numbers for 10; if we go positive, we have 10 ^{1} = 10.*1624

*Well, that is not going to work, because we are going to need a fraction.*1629

*So, we notice that 10 ^{-1} is 1/10; and then, if we think about that, 10^{-1} would be 1/10;*1632

*10 ^{-2} would be 1/100; 10^{-3} would be 1/1000; 10^{-4} would be 1/10000.*1638

*So, 10 ^{-4} = 1/10000; we can also see this, because we can count the number of 0's: 1, 2, 3, 4.*1645

*So, that is 10 ^{4}, and since it is 1/10^{4}, then that must be 10^{-4}.*1655

*We have that -4 is what we have to raise 10 to, to get 1/10000.*1661

*The natural log of e ^{17}: well, remember: natural log is just another way of saying log base e.*1668

*So, log _{e}(e^{17}): what number do we have to raise e to?*1673

*e ^{?} =...well, the thing we are working with is e^{17}, so e^{17} would be e^{?}.*1683

*Well, that is pretty clear: the thing that the question mark has to be is the 17.*1691

*Otherwise they will never match up; so it must be e ^{17} that we want here, so 17 is our answer,*1695

*because if we raise e to the 17, it is no surprise that we get e ^{17}.*1702

*Finally, log _{4}(32): once again, we are saying, "What is the number that we have to raise 4 to, to get 32?"*1707

*So, we move that over; we can think of this as 4 ^{?} = 32.*1714

*So, 32 = 4 ^{?}: well, let's start looking at some possible numbers for 4.*1720

*We could have 4 ^{1}; that would just be 4--not big enough.*1725

*4 ^{2} would be 16; we are starting to get close.*1730

*4 ^{3} would be 64--it looks like we overshot.*1733

*Well, we might notice that 16 times 2 equals 32; so if we could somehow get 2 to show up, we would be good.*1738

*Notice: how is 4 connected to 2--what is the connection between these numbers?*1746

*Well, the square root of 4 is equal to 2; but we also had another way of saying that: 4 ^{1/2} is the same thing as saying square root.*1753

*4 ^{1/2} = 2; so we see that 4^{2} times 4^{1/2} equals 32.*1762

*4 squared times 4 to the 1/2 (4 squared is 16; 4 to the 1/2 is 2)--16 times 2--gets us 32.*1770

*So now, we just need to combine those: 4 ^{2} times 4^{1/2} is just another way of saying 4^{4/2} times 4^{1/2}.*1778

*We can add them--they are on a common base; so, 4 ^{5/2}...*1790

*So, the answer for this--the number that we have to raise 4 to, to get 32, is 5/2.*1795

*All right, what if we were doing the other direction--if we wanted to write an exponential equation in logarithmic form?*1803

*We have these exponential equations: 3 ^{4} = 81--and now we want to do it in the logarithm form.*1809

*Remember, we have that log _{a}(x) = y is the same thing as saying a^{y} = x.*1814

*Remember, our base here--we can think of it as popping up under what is on the other side of the equation.*1824

*So, this over here is the exponential form; this here is the logarithmic form.*1830

*Logarithmic form is this log stuff, and exponential is this a to the something stuff.*1838

*So, what we have is exponential forms here; we want to flip it.*1843

*3 ^{4} = 81: that is going to be log...what is our base? Our base here is a 3, so log_{3}...*1847

*what is the number that we are raising to? That is the blue, so we are not going to use that.*1858

*Finally, the number that we have is x...so, log _{3}(81) = the number that we have to raise to, 4, because 3^{4} = 81.*1862

*If we ask what number we have to raise 3 to, to get 81, that is going to be 4; 3 ^{4} = 81.*1878

*We can do this with any of this stuff: 10 ^{2.4} = 251.18.*1887

*Then, that is going to be...our base is 10, so we can write that as just log, because if we don't have a base, it just says log base 10.*1894

*log of what number? Our number that we are going to get to is 251.18,*1902

*and it actually keeps going, so we will leave those dots there to show that it keeps going.*1908

*And that is going to end up equaling 2.4, because the number that we have to raise 10 to,*1915

*to get 251.18, is 2.4, as was shown to us in our original exponential form.*1921

*So, another one: our base here is e, so we can write that as natural log of this number.*1929

*We were told that it comes out to be 4.*1937

*Finally, our base is π: so log _{π}(this number) = √(1/2), because we know that,*1942

*if we raised π to the √(1/2), we would get 2.2466, and continuing on.*1955

*So, that is how we were able to figure out that log _{π}(2.2466...) must be the square root of 1/2.*1962

*Graph f(x) = log _{3}(x): to do this, we want to start with a nice table to figure out the values.*1971

*x; f(x); notice that we probably don't want to just toss in numbers right away.*1978

*If we plug in 10, well, I don't know what number we have to raise 3 to, to get 10.*1986

*That is going to be something complicated; we have to use a calculator.*1992

*But we do know what it is going to be if we plug in numbers like, say, 3.*1994

*If we plug in 3, what number do you have to raise 3 to--what is log _{3}(3)?*1998

*What number do we have to raise 3 to, to get 3?*2004

*That is easy: we just have to raise it to the 1--nothing at all.*2006

*We don't have to raise it to anything, other than what is already there; so just something to the 1 is what it starts as.*2009

*What about 9?--well, what number do we have to raise 3 to, to get 9?*2015

*We have to square it, so we have to raise it to the 2.*2019

*We can keep going in this pattern: what number do we have to raise 3 to, to get 27?*2021

*We have to raise it to the 3.*2025

*What number do we have to raise 3 to, to get 81? We have to raise it to the 4.*2027

*And we can keep going if we want.*2031

*What if we went in the other direction? Well, for 2, we don't know what number we would have to raise 3 to.*2032

*But for 1, yes, we do know what number we would have to raise 3 to.*2037

*3 to the what equals 1? Just like everything else, 3 to the 0 equals 1.*2039

*We could go to 1/3: what number do we have to raise 3 to, to get 1/3? -1.*2047

*What number do we have to raise 3 to, to get 1/9? -2.*2053

*And it would get lower and lower and lower, the closer we got to 0.*2057

*Once again, we will never actually be able to get to 0, because there is no number that we could raise 3 to, to get 0.*2060

*But we can get really, really close to 0.*2065

*So, at this point, we are ready to actually try plotting it.*2067

*Notice: our x-values go pretty widely; so let's look at x-values going from -10 up to +100.*2069

*And let's look at our y-values: our y-values, our f(x) values, don't really manage to change very much.*2078

*So, we will look at y-values only going from -3...oh, let's make it -5...up to positive 5.*2083

*OK, let's start drawing that in; we start here; here is our x-axis and y-axis.*2092

*Make a scale; the scale for the x will be in chunks of 10, because we have to cover a lot of ground: -10, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.*2103

*We can keep going if we want, but that is good enough for us.*2120

*So, here is a 10; here is 100; I will mark 50 in the middle; 1, 2, 3, 4, 5....50, and -10.*2123

*So, we can see the scale on it here.*2131

*For our verticals, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5.*2133

*Here is -1 and -5, positive 1 and positive 5.*2144

*Great; now we are ready to plot down some points.*2150

*We see a 3; we are at 1; so we are very close here; we are a little under 1/3 of the way up to the 10.*2152

*So, let's say it is about here; at 9, just a little bit before the 10, we are at 2.*2159

*At 27 (10, 20, 30--a little bit before that, but a little under 1/3 of the way towards the other side), we are at 3.*2167

*At 81 (100, 90, 80--just a hair in front of 80), we manage to be at 4.*2180

*There we go; now we want to go the other way, as well.*2190

*At 1/3...at 1, we are at 0, so we are really, really close to that y-axis already; at 1/3,*2194

*now we are getting pretty close; we are at -1 at 1/9; we are practically on top of it,*2202

*but we will never actually be on top of it; we will just get really, really close.*2206

*And we can see that this pattern is going to continue: 1/27, -3; 1/81, -4.*2210

*So, as it gets really close to the 0, it is going to just shoot down really quickly.*2216

*So, let's draw this side in; it approaches this asymptotically.*2221

*It gets really, really close, but it will never actually touch it; the part where it looks like it is touching it is just my human error at fault.*2228

*But it is not going to ever quite touch it on a perfect graph.*2235

*It might look like it, because of the thickness of the lines, but it will never actually do it.*2239

*And as it grows more and more, it slows down, because it has to go even farther out to be able to get any growth.*2242

*It slows down the farther out it gets; and we graph log _{3}(x).*2250

*Cool; finally, what are the domains of these functions? f(x) = log _{7}(-x + 2).*2254

*Remember: the idea we had was log _{a}(stuff); then this stuff here must always be positive.*2261

*So, it must be positive; otherwise, it just doesn't work.*2271

*If we try to take the log of 0, it doesn't work; if we try to take the log of a negative number, it doesn't work.*2276

*You always have to take the log of positive numbers, whatever the base is.*2282

*For any base, this is going to be the case; so it doesn't matter if it is base 7 or base fifty billion.*2286

*It is going to be the case that we have to have whatever is inside of the logarithm,*2290

*whatever the logarithm is operating upon--it has to be greater than 0; it has to be a positive number.*2294

*So, we know that the thing that log is operating on here is -x + 2.*2300

*So, we know that -x + 2 must be positive; it must be greater than 0.*2304

*We move the x over; we have that 2 has to be greater than x, so x has to be less than 2,*2308

*and it can go all the way down to negative infinity, because the only restriction we have is that 2 is greater than x,*2314

*which we could write out as...anywhere from negative infinity up until positive 2, but not including positive 2,*2318

*which we show with a parenthesis to show that it is not inclusive.*2325

*Over here, g(t) = 5t(log _{π}(3t + 7)).*2329

*Once again, the base doesn't really matter; it has to be positive, no matter what the base is.*2333

*For any arbitrary base a, it has to be positive on what the logarithm is operating on.*2339

*We look at the 5t part: we might get worried--"oh, is the 5t going to interact with it?"*2345

*5t times log _{π}...5t is really in its own world; it is doing its own thing.*2349

*5 times t...we can do that for any number; we can multiply 5 times any number, so its domain is anything at all.*2353

*It is not going to actually get in our way; once again, the only thing we are worried about is*2359

*when the logarithm is going to try to take the log of a negative or 0 number.*2362

*So, to avoid that, we have to have that 3t + 7 must be greater than 0;*2368

*otherwise, we will be taking the log of something that we cannot take logs of, that would break our function.*2373

*So, 3t + 7 is greater than 0; subtract 7; 3t > -7; divide by 3; t must be greater than -7/3.*2378

*So, t starts at -7/3, but is not actually able to include -7/3.*2388

*So it starts just above -7/3 and can go anywhere larger; so it can go all the way out to positive infinity.*2392

*So, we have -7/3 shown with a parenthesis, because we can't actually include -7/3; we can just get arbitrarily close to it.*2397

*It is going all the way out to positive infinity.*2404

*And there are our two domains; all right, cool.*2407

*We will talk a bunch more about logarithms in the next one, where we will explore the properties of logarithms.*2409

*And then, we will see even more about how the two connect.*2413

*We have a lot of really interesting ideas; it is new stuff, but once you start practicing it,*2414

*as you do it a bunch of times, logarithms will really start to make sense.*2418

*You will get this idea of "what am I trying to raise this number to, to get the thing I am taking the logarithm of?"*2420

*What does this base have to be raised to, to get the number that I am taking the log of?*2426

*All right, we will see you at Educator.com later--goodbye!*2430

2 answers

Last reply by: Duy Nguyen

Thu Aug 6, 2015 10:24 PM

Post by Duy Nguyen on August 6, 2015

Hi, would you mind explaining why the domain of a log function does not include negative numbers? Because log base (-2) of (-8) is 3 and x, in this case, is a negative number.

Thank you very much.