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Lecture Comments (3)

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.

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 (loga x), where a > 0 and a ≠ 1, is defined to be the number y such that ay = x.
    loga x = y     ⇔     ay = 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     ⇔     loge 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     ⇔     log10 x
  • 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) = loga x               f−1(x) = ax
  • Because a logarithm is the inverse of exponentiation, we cannot take the logarithm of some numbers. Exponentiation (ax) only has a range of (0,∞), so the corresponding logarithm can only reverse those same output values. In other words,
    Domain of loga x:    (0, ∞) .
  • We can also see the above domain must be true because it would make no sense otherwise. Consider that log2 0 = b means that 2b = 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

Transcription: Introduction to Logarithms

Hi--welcome back to

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 23,0008

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 log2(x), with this little 2 as a subscript right here,0055

is defined to be the number y such that 2y = x.0065

So, when we see log2(x) = y, then we know that that would be 2y = 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 log2(8) = 3, the reason why that is the case is because0094

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 23, we get 8, so we know that log2(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 log2(x) = y, you can think of this as...if the 2 were under the y,0120

if we had 2y, 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: log2(32) = 5, because if we raise 2 to the 5th, we get 32.0149

2 times 2 times 2 times 2 times 2 is equal to 32; 4, 8, 16, 32.0158

log2(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

log2(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 out0194

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, loga(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 ay = x.0224

If we take loga(x), then we know that that gives us some y, and that ay = 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 ay.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

log7(49) = 2, because, if we move this base over,0319

then 72 is just 7 times 7, which gets us 49, which is exactly what we started with.0324

So, this is the case, because 72 = 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 72, like this.0343

The same thing over here: we know that log10(10000) = 4,0350

because, if we move our base under, we know that 104 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 log10(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 104; 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 log5(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 log4(2), then we see that 41/2 is equal to 2.0433

What is 1/2? 1/2 is an exponent that means square root; 41/2 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 log1/2(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 logπ(π), then we know that it has to come out as 1,0486

because π1--of course, that is no surprise that it is going to equal what it already started with.0492

So, if we are taking logπ(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: loge(1) becomes 0, because if we move this over, e0, 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 log10(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 log7(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 loge, 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 loge(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 log10(x).0657

Well, we can find the value of expressions like log2(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 loge 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 log10 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 log3, most calculators won't have an easy way for us to just get what log3(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) = log2(x) (this is in red); g(x) = log5(x)0809

(that one is in blue), and finally h(x) = log10(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, a0, = 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, log2 is only going to be at a meager 4.0873

But for log10, when we look at log10, 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 log10, because 102 = 100.0889

We are seeing a similar thing for log5: it has to get all the way up to 25 before it hits this height of 2, as well, because it is 52.0899

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 1250909

as an input value, because 53 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 log2(x) = y, then we can see its flip of 2y = x.0966

We just change the x and the y location.0973

So, if we take log2(8), that becomes 3, because remember: 23 = 8.0975

So, when we take log2(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 23--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 log2(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) = loga(x), f-1(x) = ax.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 ax, then it is going to be...1071

well, we will apply f-1 next; first, f(x) = loga(x).1077

We have loga(x); then we apply the f-1, and we have aloga(x).1082

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

Well, remember: loga(x) = y is the same thing as saying ay = x.1094

So, that is what loga(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, loga(x) = y: we can just say, "Whatever the number loga(x) is, let's call it y."1115

So, we can swap that out, and we can say, "ay," just because we are saying we will call loga(x) y.1120

That is what we have over here; but remember, we defined this idea of what loga(x) is based on ay = x.1126

Well, we now have ay = x; so if ay = 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, 2x (that one is in red) and log2(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 log2 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 2x 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) = ax is going to be 0 to infinity.1275

ax...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 2x, 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 ax or the example 2x, 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 log2(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 2b 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

2b 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 2b = -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 log2(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 log6(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

61--that is just 6; 62--well, that would be 36; 63 is 180 + 36 = 216.1579

That is what we are looking for; so it must be the case that it is 3: log6(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, log10 (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 101 = 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 104, and since it is 1/104, 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 e17: well, remember: natural log is just another way of saying log base e.1668

So, loge(e17): what number do we have to raise e to?1673

e? =...well, the thing we are working with is e17, so e17 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 e17 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 e17.1702

Finally, log4(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 41; that would just be 4--not big enough.1725

42 would be 16; we are starting to get close.1730

43 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: 41/2 is the same thing as saying square root.1753

41/2 = 2; so we see that 42 times 41/2 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: 42 times 41/2 is just another way of saying 44/2 times 41/2.1778

We can add them--they are on a common base; so, 45/2...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: 34 = 81--and now we want to do it in the logarithm form.1809

Remember, we have that loga(x) = y is the same thing as saying ay = 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

34 = 81: that is going to be log...what is our base? Our base here is a 3, so log3...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, log3(81) = the number that we have to raise to, 4, because 34 = 81.1862

If we ask what number we have to raise 3 to, to get 81, that is going to be 4; 34 = 81.1878

We can do this with any of this stuff: 102.4 = 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 logπ(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 logπ(2.2466...) must be the square root of 1/2.1962

Graph f(x) = log3(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 log3(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/ 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 log3(x).2250

Cool; finally, what are the domains of these functions? f(x) = log7(-x + 2).2254

Remember: the idea we had was loga(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(logπ(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 logπ...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 is2359

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 later--goodbye!2430