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### Exponential Functions

- Know the graph of the exponential function and its properties.
- If the base is greater than 1, the function is exponential growth. If it is between 0 and 1, it is exponential decay.
- Solve exponential equations
*with the same base*by equating the exponents. - Solve exponential inequalities
*with the same base*by applying the same inequality to the exponents.

### Exponential Functions

^{x}. On the graph correctly identify the y - intercept.

- Create a Table of values in order to graph the Exponential Function.
x f(x) = 3*( [1/2] ) ^{x}-2 3*( [1/2] ) ^{ − 2}= 3*( [2/1] )^{2}= 12-1 3*( [1/2] ) ^{ − 1}= 3*( [2/1] )^{1}= 60 3*( [1/2] ) ^{0}= 3*1 = 31 3*( [1/2] ) ^{1}= [3/2]2 3*( [1/2] ) ^{2}= [3/4]3 3*( [1/2] ) ^{3}= [3/8]- Draw a smooth curve. Your graph should never touch the x - axis.

^{x}. On the graph correctly identify the y - intercept.

- Create a Table of values in order to graph the Exponential Function.
x f(x) = 4*2 ^{x}-6 4*( 2 ) ^{ − 6}= 4*( [1/2] )^{6}= [4/64] = [1/16]-4 4*( 2 ) ^{ − 4}= 4*( [1/2] )^{4}= [4/16] = [1/4]-2 4*( 2 ) ^{ − 2}= 4*( [1/2] )^{2}= [4/4] = 10 4*( 2 ) ^{0}= 4*1 = 42 4*( 2 ) ^{2}= 4*4 = 164 4*( 2 ) ^{4}= 4*16 = 64- Draw a smooth curve. Your graph should never touch the x - axis.

f(x) = 7*( [1/4] )

^{ − x}

- Rewrite the equation in standard form :f(x) = a*b
^{x} - Remember that a
^{ − n}= [1/(a^{n})] - Rewritten in standard form, the function becomes f(x) = 7*4
^{x}

f(x) = 3*(6)

^{ − x}

- Rewrite the equation in standard form :f(x) = a*b
^{x} - Remember that a
^{ − n}= [1/(a^{n})] - Rewritten in standard form, the function becomes f(x) = 3*( [1/6] )
^{x}

^{m − 3}= 25

^{2m + 3}

- In order to solve for m, we need to have the same base. Rewrite problem in base 5.
- Rewrite [1/5] as base 5 =
- Rewrite 25 as base 5 =
- That would give you [1/5] = (5)
^{ − 1}and 25 = 5^{2} - Rewrite the problem.
- (5
^{ − 1})^{m − 3}= (5^{2})^{2m + 3} - Distribute the exponets.
- 5
^{ − m + 3}= 5^{4m + 6} - Now that you have the problem in the same base, eliminate the base and solve for m
- − m + 3 = 4m + 6

^{ − x}= ( [1/7] )

^{3x}

- In order to solve for m, we need to have the same base. Rewrite problem in base 7.
- Rewrite [1/7] as base 7 =
- Rewrite 343 as base 7 =
- That would give you [1/7] = (7)
^{ − 1}and 343 = 7^{3} - Rewrite the problem.
- (7
^{3})^{ − x}= (7^{ − 1})^{3x} - Distribute the exponets.
- 7
^{ − 3x}= 7^{ − 3x} - Now that you have the problem in the same base, eliminate the base and solve for x
- − 3x = − 3x

^{ − x}= 216

^{ − 2x − 1}

- In order to solve for x, we need to have the same base.
- The only base that seems to work will be base 6.
- Rewrite 36 as base 6 =
- Rewrite 216 as base 6 =
- That would give you 36 = (6)
^{2}and 216 = (6)^{3} - Rewrite the problem.
- (6
^{2})^{ − x}= (6^{3})^{ − 2x − 1} - Distribute the exponets.
- 6
^{ − 2x}= 6^{ − 6x − 3} - Now that you have the problem in the same base, eliminate the base and solve for x
- − 2x = − 6x − 3

^{ − 3x}= 25

^{3x − 3}

- In order to solve for x, we need to have the same base.
- The only base that seems to work will be base 5.
- Rewrite 625 as base 5 =
- Rewrite 25 as base 5 =
- That would give you 625 = (5)
^{4}and 25 = (5)^{2} - Rewrite the problem.
- (5
^{4})^{ − 3x}= (5^{2})^{3x − 3} - Distribute the exponets.
- 5
^{ − 12x}= 5^{6x − 6} - Now that you have the problem in the same base, eliminate the base and solve for x
- − 12x = 6x − 6

^{ − 3x}< 16

- In order to solve for x, we need to have the same base.
- The only base that seems to work will be base 4.
- Rewrite 64 as base 4 =
- Rewrite 16 as base 4 =
- That would give you 64 = (4)
^{3}and 16 = (4)^{2} - Rewrite the problem.
- (4
^{3})^{ − 3x}< 4^{2} - Distribute the exponets.
- 6
^{ − 9x}< 4^{2} - Now that you have the problem in the same base, eliminate the base and solve for x
- − 9x < 2
- Divide both sides by − 9. Remember that whenever you divide by a negative, the inequality must be switched.

^{ − 3x + 2}> 4

^{ − 2x}

- In order to solve for x, we need to have the same base.
- The only base that seems to work will be base 4.
- Rewrite 16 as base 4 =
- That would give you 16 = (4
^{2}) - Rewrite the problem.
- (4
^{2})^{ − 3x + 2}> 4^{ − 2x} - Distribute the exponets.
- 4
^{ − 6x + 4}> 4^{ − 2x} - Now that you have the problem in the same base, eliminate the base and solve for x.
- − 6x + 4 >− 2x
- − 4x + 4 > 0
- − 4x >− 4
- Divide both sides by − 4. Remember to switch the inequality whenever you divide by a negative.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Exponential Functions

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
- What is an Exponential Function?
- Graphing Exponential Functions
- Properties
- Continuous and One to One
- Domain is All Real Numbers
- X-Axis Asymptote
- Y-Intercept
- Reflection Across Y-Axis
- Growth and Decay
- Equations
- Inequalities
- Example 1: Graph Exponential Function
- Example 2: Growth or Decay
- Example 3: Exponential Equation
- Example 4: Exponential Inequality

- Intro 0:00
- What is an Exponential Function? 0:12
- Restriction on b
- Base
- Example: Exponents as Bases
- Variables as Exponents
- Example: Exponential Function
- Graphing Exponential Functions 2:33
- Example: Using Table
- Properties 11:52
- Continuous and One to One
- Domain is All Real Numbers
- X-Axis Asymptote
- Y-Intercept
- Reflection Across Y-Axis
- Growth and Decay 15:06
- Exponential Growth
- Real Life Examples
- Example: Growth
- Example: Decay
- Real Life Examples
- Equations 17:32
- Bases are Same
- Examples: Variables as Exponents
- Inequalities 21:29
- Property
- Example: Inequality
- Example 1: Graph Exponential Function 24:05
- Example 2: Growth or Decay 27:50
- Example 3: Exponential Equation 29:31
- Example 4: Exponential Inequality 32:54

### Algebra 2

### Transcription: Exponential Functions

*Welcome to Educator.com.*0000

*Today begins the first in a series of lectures on exponential and logarithmic relations, starting out with exponential functions,*0003

*beginning with the definition: What is an exponential function?*0012

*Well, an exponential function is a function of the form f(x) = a times b ^{x}, where a is not 0,*0016

*because if a were to equal 0, this would all just drop out, and you wouldn't have a function.*0027

*b is greater than 0: we are restricting this definition to values of b that are positive; and b does not equal 1.*0032

*If b were to equal 1, no matter what you made x, that would still remain 1.*0039

*And then, you wouldn't really have a very interesting function.*0044

*So, the base of the function here is b.*0047

*Now, recall earlier on, when we worked with functions and equations, we have seen things like this: x ^{2} + 2x - 1...f(x) equals this.*0050

*Here, the base was a variable, and here the exponent was a constant.*0062

*Now, we are going the other way around: in this case, with exponential functions, we are going to be working with situations*0070

*where the base is a number (a constant) and the exponent contains a variable.*0084

*And that is what makes these functions fundamentally different from some of the other functions that we have seen so far.*0101

*Examples would be something like f(x) = 6x: here the base is 6, the coefficient is just 1 (a = 1), and the exponent is x.*0110

*Or you could have something a little bit more complicated: f(x) = (3 times 1/2) ^{4x - 2}.*0128

*I have an algebraic expression, not just a single variable, as the coefficient.*0138

*So here, I have a base equal to 1/2; the coefficient is 3; and then the exponent is 4x - 2.*0142

*Starting out by looking at the graphs of exponential functions: as we have done with other types of functions,*0153

*we can use a table of values to graph an exponential function.*0160

*And we are going to look at a few different permutations of these functions and see what we end up with.*0163

*I am going to start out with letting f(x) equal 3 ^{x}; let's find some values for x and y.*0170

*I am going to just rewrite f(x) as y.*0180

*If x is -3, then what is y? Well, this is giving me 3 to the -3 power.*0190

*Recall that a ^{-n} equals 1/a^{n}; so what this is really saying is 1/3^{3}, which equals 1/27.*0197

*At this point, if you are not really comfortable with exponents and the rules and properties governing working with exponents,*0215

*you should go back and review the earlier lecture on that, because we are going on to solve equations using these.*0220

*So, you need to have the rules learned, as we will be applying them frequently.*0227

*When x is -2, this is going to give me 3 ^{-2}, which is 1/3^{2}, or 1/9.*0231

*And we continue on: when x is -1, that is 3 ^{-1} = 1/3^{1}, or 1/3.*0239

*Then, getting back to some more familiar territory: when x is 0, this gives me 3 ^{0}.*0248

*A number or variable, or anything, to the 0 power, is going to give me 1; when x is 1, f(x), or y, is 3.*0257

*When x is 2, that is 3 ^{2} is 9; when x is 3, that is 3^{3}, to give me 27.*0266

*Let's go ahead and plot these values out; let's do that so that this is -2, -4, -6, -8, so we can look at more values right on this graph.*0276

*2, 4, 6, 8, 2, 4, 6, 8, 10; -2, 4, 6, 8, 10; so -10 is down here.*0288

*When x is -3, the graph is just slightly above 1; it is 1/27; when x is -2, we get a little farther away from 1--it becomes 1/9.*0299

*When x is -1, the graph rises up even a little more and becomes -1/3.*0312

*When x is 0, y is 1; when x is 1, y is 3; when x is 2, y is right up here at 9; and then, when x is 3, y gets very large.*0321

*This gives me a sense of the shape of the graph--that as x becomes positive, y rapidly becomes a very large number;*0336

*as x is negative, what I can see happening is that f(x) is approaching 0, but it never quite gets there.*0345

*Therefore, what I have is that, for this graph, the x-axis is a horizontal asymptote.*0358

*So, this is the graph of f(x); let's look at a different case--let's look at the graph of...this is in the form f(x) = ab ^{x};*0377

*let's look at the graph f(x) = a(1/b) ^{x}.*0392

*Well, a is just 1, so I am going to look at the situation g(x) = 1 over 3 to the x power.*0398

*I am making a table of values, again, for x and y.*0410

*Again, when x is -3, let's look at what y is; it is going to be 1/3 to the -3.*0419

*Using this property, I am then going to get 3 ^{3}, which is 27.*0426

*For -2, I am going to get (1/3) ^{-2}, or 3^{2}, which is 9.*0433

*-1 is going to give me (1/3) ^{-1} = 3; 0...anything to the 0 power is simply going to be 1.*0440

*(1/3) ^{1} is 1/3; (1/3)^{2} is 1/9; and (1/3)^{3} is 127.*0453

*Let's see what happens with this graph: here, when x is -3, y is very large--it is some value way up there, 27 (that is off my graph).*0469

*Let's look at -2: when x is -2, y is 9 (that is right here).*0479

*And I know that, when I get out to -3, this is increasing; so I know the shape up here.*0484

*When x is -1, y is 3; when x is 0, y is 1; they have the same y-intercept, right here at (0,1).*0489

*When x is 1, y is 1/3; when x is 2, we see that the graph is approaching the x-axis, because now, we are here at 1/9.*0502

*And when x is 3, the value of the function is 127.*0515

*So again, I see this x-axis, again, being the horizontal asymptote for both graphs.*0519

*I know that y is getting very large as x is becoming negative.*0532

*And I know that, as x is positive, the graph is approaching, but not reaching, the x-axis.*0538

*Here, the y-axis also forms an axis of symmetry; so these two graphs are mirror images of each other.*0547

*f(x) and g(x) are mirror images reflected across the y-axis.*0553

*Let's look at one other case: let's look at the case...let's call it h(x) = 3 ^{x}, but let's take the opposite of that.*0558

*Therefore, this will be simpler to figure out the values: let's just leave the x-values the same as they were for f(x),*0569

*and let's just extend this graph out: these were my values for f(x), and now let's figure out what h(x) is going to give me.*0581

*Well, if x is -3, and I put a -3 in here, again, I am going to get 1/27, but I want the opposite of that; so I am going to get -1/27.*0592

*If x is -2, again, I am going to get 3 ^{-2}, which is 1/9, but I am going to take the opposite.*0602

*So, all I have to do is change the signs on these to get my h(x) values.*0608

*And then, we can see what this graph looks like.*0615

*This is f(x); I have g(x) here; for h(x), when x is -3, h(x) is right here; it is very close to the x-axis, but not quite reaching it.*0620

*And at -2, it is a little bit farther away at -1/9; at -1, we are down here at -1/3; at 0, here we have the y-intercept at (0,-1).*0633

*When x is 1, y is -3, right about here; when x is 2, y is down here at -9; and when x is 3, we are going to be way down here at -27.*0649

*So, for h(x), what I am going to have (I'll clean this up just a bit)...*0664

*This is the graph of h(x), and again, I am seeing the x-axis here, acting as a horizontal asymptote.*0686

*And I am seeing that, as x becomes positive, y becomes very large, but this time it is in the negative direction.*0699

*It is giving me a mirror image here with f(x); but now, my values are going down in the negative direction.*0706

*So, these are several graphs of exponential functions; let's go ahead and sum up the properties of these functions.*0712

*What we saw is that f(x) is continuous and one-to-one.*0721

*I am just very briefly sketching these three situations: I let f(x) equal 3x, g(x) equal (1/3) ^{x}, and h(x) equal (-3)^{x}.*0724

*So, for the graph of f(x), what I got is this...actually, rising faster; so let's make that rise much faster.*0746

*And I saw that it is continuous; there weren't any gaps or discontinuities.*0759

*When we worked with graphing some rational functions, we saw that there were actually discontinuities; there aren't any here.*0765

*It is also one-to-one; and I could use the vertical line test.*0771

*Recall that, if you draw a vertical line across the curve of a graph, if it only crosses the graph once,*0774

*at one point, everywhere you possibly could try, you have a one-to-one relationship; you have a function.*0782

*So, no matter where I drew a vertical line, I would only cross this curve one time.*0790

*The domain of f(x) is all real numbers: you see that I could make x a negative value; I could make it a positive value.*0795

*However, the range is either all positive real numbers or all negative real numbers.*0802

*Here, the range is positive real numbers; for h(x), I graphed that, and that turned out like this.*0807

*Here, the domain was all real numbers, but the range was the negative real numbers.*0820

*We saw that the x-axis acts as an asymptote that the graphs of these functions approach, but never reach.*0835

*The y-intercept is at (0,a); so for both g(x) and f(x), here a = 1, so the coefficient is 1; so the y-intercept is at (0,1).*0842

*However, for h(x), a = -1; the coefficient is -1, so I have a y-intercept here at (0,-1).*0861

*Finally, we saw that the graphs of f(x) = ab ^{x} and f(x) = a/(1/b^{x}) are reflections across the y-axis.*0872

*So, that was this graph, which shares the same y-intercept, but is reflected here across the y-axis.*0880

*So, looking at these three different situations and their graphs can tell us a lot about what is going on with exponential functions.*0893

*Introducing the concept of growth and decay: if you have a function in this form, f(x) = ab ^{x},*0907

*and a is greater than 0 (it is a positive number), and b is greater than 1, then this represents exponential growth.*0916

*And the concepts of exponential growth and decay are very frequently used in real-world applications.*0925

*So, we are going to delve into this topic in greater depth in a separate lecture.*0931

*But thinking about exponential growth: that could be something like working in finance, or thinking about your own savings and compound interest.*0937

*That works in such a way that the growth is exponential.*0947

*An example of exponential growth would be something like f(x) = 1/4(2 ^{x}).*0952

*And that is because here, b is greater than 1; so this is growth.*0966

*If I had another function, f(x) = 4((1/8) ^{x}), here this is representing exponential decay,*0972

*because b is greater than 0, but it is less than 1: it is a fraction between 0 and 1.*0983

*And we sometimes talk about things such as radioactive decay and half-life in terms of an exponential function.*0990

*Another way to think of this for a minute is: recall that a ^{-n} = 1/a^{n}.*0999

*So, I can actually rewrite this function as 4(8 ^{-x}).*1006

*If I had a base here that was greater than 1, but the exponent was negative, then I also know that I have decay.*1014

*If you write it in the standard form where the exponent is positive, then all you need to do is look and see the value of b.*1022

*If it is greater than 1, you have growth; if it is between 0 and 1, you have decay.*1029

*If it is a number greater than 1, and it is not in standard form, and you see I have a negative exponent,*1033

*that also gives you a clue that you are looking at decay.*1039

*But you can always put them in standard form, like this, and then just go ahead and look at the value of the base.*1043

*Working with exponential equations: in exponential equations, variables occur as exponents.*1053

*That is what I mentioned in the beginning, when I was talking about exponential expressions and exponential functions.*1059

*But now, we are talking about equations: again, you are used to working with things where the base may be a variable, but the exponent is a number.*1064

*And now, we are going to change that around and actually have situations where variables are exponents.*1072

*There are some properties of this that we can use to solve these equations.*1079

*If the bases are the same (if b ^{x} equals b^{y}), then x must equal y.*1083

*If the bases are the same, in order for the left half of the equation to equal the right half, the exponents have to be the same; they have to be equivalent.*1090

*Look at this in a very simple case: 6 ^{3x - 5} = 6^{7}.*1100

*6 is the same base as 6; so for this left half to equal the right half, if the bases are equal, these two must be equal.*1108

*So, in order to solve these, I am just going to take 3x - 5 and put that equal to 7, which leaves me with a simple linear equation to solve.*1116

*I am going to add 5 to both sides, which is going to give me 3x = 12.*1125

*Then, I am going to divide both sides by 3 to get x = 4.*1129

*Things get a little more complicated if the bases are not the same.*1137

*If the bases are not the same, the first thing to do is try to make them the same.*1141

*If that is not done reasonably simply, then we can use another technique that we are going to learn about in a later lecture.*1145

*But for right now, we are going to stick to situations where either the bases are the same, or you can pretty easily make them the same.*1152

*For example, I could be given an exponential equation 2 ^{x - 3} = 4.*1159

*These bases are not the same, so I can't use this technique.*1168

*However, you can pretty easily see that you can make them the same base*1171

*by saying, "OK, 2 squared is 4; so instead of writing this as 4, I am going to write it this way."*1175

*Now, I am back to the situation where the bases are the same; I am going to set the exponents*1185

*equal to each other (because they must be equal to each other) and solve for x: x = 5.*1191

*You could have a little bit more complicated situation, where 3 ^{x - 1} (a separate example) = 1/9.*1199

*And I can see that I want these to be the same, and I know that 1/3 squared is 1/9.*1208

*So, I am pretty close; but I need this to be 3.*1214

*Well, recall that I could rewrite this as 3 ^{-2}: 3^{-2} is the same as 1/3 squared.*1217

*This also equals 1/9; and it is fine that this is a negative exponent--I can go ahead and use it up here, rewriting 1/9 as 3 ^{-2}.*1227

*Now, -x - 1 = -2, so x = -1.*1236

*OK, so basically, when you are working with exponential equations,*1252

*if they are the same base, you simply set the exponents equal to each other,*1259

*because this property tells us that that must be the case.*1268

*If the bases are not the same, try to make them the same: that is going to be your first approach.*1272

*And then, once you have written them as the same base, then you go ahead and solve by setting the exponents equal to each other.*1278

*Now, let's look at exponential inequalities: exponential inequalities involve exponential functions.*1289

*We just talked about exponential equations, and this is a similar situation, except we are working with an inequality, not an equation*1296

*(greater than, less than, greater than or equal to, less than or equal to).*1303

*And there are some properties that we can use to help us solve these.*1307

*If b > 1, then b ^{x} > b^{y} if and only if x > y.*1311

*And this makes sense: if I have the same base (these are the same),*1320

*the only way that this left half is going to be greater than the right half*1325

*is if these exponents hold the same relationship, where x is greater than y.*1330

*And this could be greater than or equal to, or less than, or less than or equal to.*1335

*So, b ^{x} < b^{y} if and only if x < y--the same idea.*1339

*If the bases are the same, and this on the left is less than the one on the right,*1346

*then that relationship, x < y, must be holding up.*1350

*We are going to use this property to solve inequalities.*1355

*For example, 4 ^{x + 3} > 4^{2}: I know that the bases are equal, so that x + 3 must be greater than 2.*1357

*So, I just solve, subtracting 3 from both sides to get x > -1.*1371

*Again, if you are trying to work with a situation where you are solving an exponential equation or inequality,*1378

*and the bases aren't the same, see if you can make them the same.*1384

*5 ^{6x} < 115 well, I know that 5^{2} is 25; if I multiply 25 times 5, I am going to get 125.*1389

*Therefore, 5 ^{2} times 5...that is 5^{3}...equals 125.*1404

*I am going to rewrite this as 5 ^{3}; now that the bases are the same, I can say, "OK, 6x is less than 3, so x is less than 1/2."*1410

*Again, the idea is to get these in the form where the base is the same,*1426

*and then use the property that the relationship between the exponents has to be maintained,*1431

*according to the inequality, if the bases are the same.*1439

*Looking at examples: let's go back to talking about graphing.*1446

*We are asked to graph this function, f(x) = 3(2 ^{x})--graphing an exponential function.*1449

*x...and we need to find y, so that we can do some graphing.*1457

*When x is -3, this is going to be 3 times 2 to the -3 power.*1467

*Again, this is equal to 3 times 1 over 2 ^{3}; this is going to be equal to 3 times...2 times 2 is 4, times 2 is 8; or 3/8.*1474

*When x is -2, I get 3 times 2 ^{-2}; 3 times 1/2 squared...2 times 2 is 4, so this gives me 3 times 1/4, or 3/4.*1492

*-1: 3 times 2 ^{-1} equals 3 times 1/2^{1}...remember that this -1 tells us that we need to take 1/the first power.*1509

*So, that is 3 times 1/2, which is 3/2.*1530

*0 gives me 3 times 2 ^{0}, or 3 times 1, which equals 3.*1537

*Using 1 as the x-value gives me 3 times 2 ^{1}, or 3 times 2, which equals 6.*1546

*Using 2 gives me 3 times 2 squared, which is 3 times 4, or 12.*1555

*And then, one more: 3 times 2 cubed equals 3 times 8, or 24.*1562

*Plotting these values: when x is -3, y is 3/8 (just a little bit greater than 0); when x is -2, y gets a little bit bigger; it is 3/4.*1570

*When x is -1, then y becomes 3/2, or 1 and 1/2; 3/2 is going to be about here.*1596

*When x is 0, y is 3; when x is 1, y is up here at 6, rising rapidly; when x is 2, y will be all the way up here at 12.*1607

*I have enough points to form my curve; and I see the typical properties of exponential functions.*1620

*Remember that the y-intercept is going to be at (0,a): in this case, a = 3, so my y-intercept is going to be at (0,3), and that is exactly what I see.*1629

*And I can see that over here in the table of values.*1646

*I also see that the x-axis is an asymptote, and that the graph is approaching, but never reaching, the x-axis.*1648

*And I also can see that, as x becomes large in the positive direction, the value of y rapidly increases.*1659

*This is a graph of a typical exponential function.*1667

*Example 2: Does this function represent exponential growth or decay?*1671

*Recall that, if a is greater than 0, then we can look at a function in the form f(x) = ab ^{x} and evaluate b.*1677

*If the base is greater than 1, then we have exponential growth.*1691

*If b is greater than 0, but less than 1, we have exponential decay.*1698

*But the caveat is that it has to be in the standard form; and I have an equation here that is not in standard form.*1704

*But I can use my rule that a ^{-n} equals 1/a^{n}.*1711

*Therefore, I am going to rewrite this as f(x) = 4((1/5) ^{x}).*1716

*So, I just took the reciprocal of the base and rewrote it.*1727

*And this is talking about a different 'a' than I am talking about here, with the coefficient: here, the coefficient is a.*1735

*All right, now that I have it in this form, I can evaluate; and I see that my coefficient is greater than 0, and that b is actually between 0 and 1.*1742

*Therefore, this is exponential decay.*1751

*I also could have looked back at the original and recalled that, if b is greater than 1,*1756

*and then you have a negative exponent, that would also give me a clue that this is exponential decay.*1760

*But one way to go about it is just to put it in the standard form, and then evaluate the value of the base.*1765

*All right, in Example 3, I have an exponential equation; and recall that, if the bases are the same,*1773

*then the exponents must be equal in order for the equation to be true.*1779

*The problem is that the bases aren't the same; so I need to get them to be the same.*1787

*Looking at these, they are both even; so I am going to try 2.*1793

*And we know that 2 cubed is 8; so let's start from there: 8 times 2 is 16; 16 times 2 is 32, so this gives me 16, 32...*1796

*times 2 is 64; times 2 is 128; so 2 to the third, fourth, fifth, sixth, seventh, equals 128.*1817

*If 2 ^{7} is 128, then 2^{-7} = 1/128.*1831

*So, I have this written as a power of 2; let's look at the right.*1839

*I know that 2 ^{7} is 128; 128 times 2 is 256; 2^{7} times 2^{1}...*1844

*I add the exponents, so this is actually 2 ^{8} = 256.*1856

*Now, I can write this equation using the base of 2.*1863

*On the left, I am going to have 2 ^{-7} raised to the 2x power, equals...*1867

*and then, on the left, 2 ^{8} times 3x - 1.*1875

*So, this is going to give me 2 to the -7 times 2x (is going to be -14x), equals 2 to the 8, times 3x - 1.*1883

*Now, I have it in this form, where the bases are the same, and I have an x and a y.*1897

*Therefore, -14x = 8(3x - 1); and then I am just left with a simple linear equation.*1901

*-14x = 24x - 8; I am going to add 14x to both sides to get 38x (I added 14 to both sides).*1911

*At the same time, I am going to add 8 to both sides to get it over here.*1926

*I could have done it the other way; then I would have had a negative and a negative, and then divided or multiplied by -1.*1930

*Anyway, I am going to go ahead and divide both sides by 38; and this is going to give me x = 8/38.*1936

*I can do a little simplification, because I have a common factor of 2.*1949

*I can cancel that out to get 2/19.*1952

*The hardest part for solving this was simply getting them into the same base.*1956

*And I was able to do that because 2 ^{-7} is 1/128, and 2^{8} is 256.*1961

*Once I did that, it was simply a matter of putting the exponents equal and solving a linear equation.*1967

*Here we have an exponential inequality: again, the first step is to get the bases the same,*1976

*because I know that, if I have a base raised to a certain power, and it is less than that same base*1981

*raised to another power, that the relationship between these two exponents has to hold up.*1992

*Well, this time (last time I had some even numbers; I tried 2, hoping I could find a base there*2001

*that I could easily make them into a common base, and I did), since I see a 27, I am going to work with 3's.*2008

*So, I know that 3 cubed is 27; therefore, 3 ^{-3} = 1/27.*2015

*243...let's look at that: if 3 ^{3} is 27, I multiply that times 3; that is going to give me 81.*2028

*If I multiply that by 3, fortunately, I am going to get 243.*2037

*So, this is 3 ^{3} times 3; that is 3^{4}, times 3, so this gives me 3^{5} = 243.*2044

*Let's go ahead and work on this, then: rewrite this as, instead of 243...I am going to write it right here as 3 ^{5},*2052

*raised to 4x - 3, is less than...instead of 1/27, I am going to write this as 3 ^{-3}, all to 6x - 4.*2063

*So, this is going to give me 3 to the 5, times 4x - 3, is less than 3 to the -3 times 6x - 4.*2074

*Now, I have the same bases: I can just look at the relationship between the exponents:*2087

*that 5(4x - 3) is less than -3(6x - 4), and solve this linear equation.*2092

*This gives me 20x - 15 < -18x + 12; that is 38x - 15 (I added 18 to both sides) < 12.*2101

*So, 38x (I am going to add 15 to both sides to get this) < 27.*2116

*And then, I am just going to divide both sides by 38 to get x < 27/38.*2123

*Again, the most difficult step was just getting these to be written as the same base.*2129

*And I was able to do that by using 3 as the base.*2135

*And then, I had equivalent bases; I did a little simplifying, set them equal, and solved this linear inequality.*2140

*That concludes this lesson of Educator.com on exponential equations and inequalities; thanks for visiting!*2150

1 answer

Last reply by: Dr Carleen Eaton

Mon Aug 8, 2016 8:40 PM

Post by Matthew Johnston on August 8, 2016

In example 3 shouldn't 8/38=4/19 instead of 2/19?

0 answers

Post by Krishna Vempati on April 6, 2015

I watched this lecture and I tried solving a problem in math class but I had trouble here is the problem.....

Your parents offer to pay you exponentially to study for your Algebra test. They say that if you study for one hour you'll get $6, two hours gets you a total $7, three hours $9, four hours $13, etc. What equation are they using to come up with those values?

Could you please help me solve this?

1 answer

Last reply by: Dr Carleen Eaton

Sun Mar 15, 2015 11:20 PM

Post by Daija Jenkins on March 9, 2015

Directions: Write an exponential function for the graph that passes through the given points.

How do I solve: (0,-5)and (-3, -135)

1 answer

Last reply by: Dr Carleen Eaton

Sat Sep 14, 2013 2:58 PM

Post by Tami Cummins on August 27, 2013

Isn't negative 3 raised to a negative 2 power still a positive 1/9. When you square the negative 3 doesn't it become positive?

1 answer

Last reply by: Dr Carleen Eaton

Tue Jul 3, 2012 7:24 PM

Post by Laura Gilchrist on June 27, 2012

If there is no variable in the exponent, will it just be a power function instead? Does it have to have variable for it to be exponential? Thanks!!

1 answer

Last reply by: Dr Carleen Eaton

Mon Apr 16, 2012 10:18 PM

Post by Ed Grommet on April 13, 2012

FOr some reason it will not play. Question is IF i have a expo equations of y=-5^x same as y=(-1)(5^x) ? Also is it decay or growth since it is not above the x axis?

1 answer

Last reply by: Dr Carleen Eaton

Mon Mar 19, 2012 3:51 PM

Post by Ding Ye on March 19, 2012

This is a really nice video. Thanks a lot!

1 answer

Last reply by: Dr Carleen Eaton

Thu Jan 26, 2012 7:53 PM

Post by Jose Gonzalez-Gigato on January 24, 2012

In the slide labeled 'Properties', at about 12:50, you mention f(x) is 'one-to'one' and give the reason that it passes the vertical line test. For a function to be 'one-to-one' it must pass the horizontal line test.

1 answer

Last reply by: Dr Carleen Eaton

Wed Jan 11, 2012 12:38 AM

Post by Arlene Francis on January 9, 2012

Are there extra examples of problems.

1 answer

Last reply by: Dr Carleen Eaton

Wed Dec 28, 2011 9:10 PM

Post by Jonathan Taylor on December 27, 2011

Dr carleen must the base be the same in all exponential equation are is this only when your working with certain exponential fuction

0 answers

Post by Guillermo Marin on August 8, 2010

Dr. Eaton is really OUTSTANDING!

0 answers

Post by Dr Carleen Eaton on May 18, 2010

Correction to Example III: The solution, x = 8/38 reduces to 4/19, not 2/19