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### Course: Precalculus (Eureka Math/EngageNY) > Unit 3

Lesson 15: Topic C: Lessons 20-21: Inverse relationship of exponentials and logarithms- Relationship between exponentials & logarithms
- Relationship between exponentials & logarithms: graphs
- Relationship between exponentials & logarithms: tables
- Relationship between exponentials & logarithms
- Half-life and carbon dating
- Exponential model word problem: medication dissolve
- Exponential model word problem: bacteria growth
- Exponential model word problems

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# Relationship between exponentials & logarithms: graphs

Given a few points on the graph of an exponential function, Sal plots the corresponding points on the graph of the corresponding logarithmic function. Created by Sal Khan.

## Want to join the conversation?

- Hi,

This might be a silly question but how is y = log_b (X) the inverse of y=b^x? Isn't the inverse of y=b^x this log_b (y) = X?

Thanks(172 votes)- I guess if you look at the function as y = f(x) = b^x. The inverse of the function would be x = f(y) = b^y. So, the logarithmic form of the inverse function is log_b(x) = y.

Ey that's just my take, might be wrong too lol.(20 votes)

- How did he know to use points 1,4, and 16 in the second chart? I thought that, given the question, he would have taken the x values from the existing points and plugged them into the 2nd function.(47 votes)
- For anyone who watches this video and has a similar question, think about it this way:

y = b^x can be rewritten as log_b(y) = x.

So for b = 4 (you can deduce that from the graph) and y = 1, 4, 16 (also deduced from the graph) you have

log_4(1) = 0

log_4(4) = 1

log_4(16) = 2

Which represent points (1, 0), (4, 1) and (16,2) the inverse of the points on the given graph!(49 votes)

- at around1:40, why is sal taking the y values of the first table and using them as the x values of the second table?(12 votes)
- I had a little trouble on this too, but here's what I figured: The
**purpose of an exponential function is to find what value a number equals when it is raised to the power of x**. However, the *purpose of a logarithm is to find the exponent (x value)* that gives you a certain amount when you raise a base by the unknown x value. Though it might sound complicated, if you try to define the purpose yourself, you'll end up with the same answer. For this reason, they are like the inverse functions of each other, just like multiplication and division.

In other words, the logarithm tries to lead you to the exponent needed to reach the value, while the exponential graph tries to lead you to the value given by the exponent's use.

Therefore, they are inverse operations as they undo each other. Furthermore, they reflect over the y=x line, and their coordinates are switched. If you check at3:29, you'll see Sal agrees. If you check the background at this time, you'll also see how when the b= 4, everything fits together perfectly.

Hope this helps.(12 votes)

- Sorry about this but I have two questions. The first might be silly, but here goes: How does one solve for x when it is an exponent such as 2^x=1/64. I've been having to guess this whole time and try for answers on my calculator, because solving for x has not been working. I think I might be doing it wrong...

Second is, I didn't understand this video too well. I don't understand what the graph has

to do with any of this, even though I've been okay with everything pertaining to logarithms so far. Any answer is much appreciated. Thank you!(6 votes)- When the variable is in the exponent, you need to use logarithms of whatever the base of the exponent is.

For 2^x = 1 / 64, the base is 2. Therefore, we'll be taking log base 2 of each side of the equation.

But before doing that, it's usually easiest to express both sides of the equation using the same base.

So, 2^x = 1 / 64 = 1 / 2^6 = 2^(-6)

log_2 ( 2^x ) = log_2 ( 2^-6 )

x = - 6

Hope this helps.

(I'll let someone else pick up on your question about the graph.)(10 votes)

- why did he take the y values and plug them into the log function? How did he know not to just use the same x values for the log function as for the exponential function?(11 votes)
- This is because logarithmic functions and exponential functions are inverse functions of each other. The inverse of a function is the function that takes the range (y) of that function to the domain (x) of the function. Therefore, the domain (x) of the inverse function will be the range (y) of the original function and the range (y) of the inverse function will be the domain (x) of the original function.

y=b^x:

Domain: Range:

0 -----> 1

1 -----> 4

2 -----> 16

log_b(x)=y

Domain: Range:

1 -----> 0

4 -----> 1

16 -----> 2

See how the logarithmic function's domain is the exponential function's range and the logarithmic function's range is the exponential function's domain. In a simpler way of thinking about it, you swap the domain and range, or x and y. This is also why with a function with the point (x,y), the inverse of the function will have the point (y,x).(1 vote)

- Hi, can you please explain how did you know that y=log_b(x) is inverse of y=b^x. a detailed answer would be appreciated. Thanks.(6 votes)
- The actual definition of a logarithm that proves this is several years ahead of the math you're having now, so it wouldn't mean much to you at this stage.

But, the short answer is that the logarithm was invented to be the inverse operation of an exponential function. In other words, there is a difficult mathematical process that was worked out over the course of about 80 years by Nicholas Mercator, Leonhard Euler and others. The term "log" is just a shorthand way of expressing this difficult bit of math.

So, a log is just a quick way of writing the inverse function of an exponential function. Thus, by definition, the log must be the inverse function of the exponential function.

So, the reason you are not given exactly what a log is or how to compute it by hand at this level of study is that it is very difficult math. Since we have calculators that can do this math for us, it is possible for people to use logs without having to master very advanced and difficult computations.

But, just for reference, here is one way to write the actual definition of a logarithm

log_b (x)= lim n→0 [x^(n) - 1] / [b^(n) - 1]

where b is the base of the logarithm.(8 votes)

- At0:44, how do we know that b raised to the power 1 is equal to four. Is it a definite variable with its value(7 votes)
- It is a constant that we deduced from the graph that is given to us.(6 votes)

- I'm sorry but I'm confused by this video. I've understood everything so far but I got lost. How does this work?(7 votes)
- You have your input x in the Log function, and the output is graphed.

Like when x=2 on the equation and say it equals 5 when x=2

you would graph 2,5(3 votes)

- To solve for other points, couldn't I figure out that b=4 from the pattern in y=b^x and go from there?(5 votes)
- From this set it is obvious that b=4, but the point of the exercise is to highlight the fact that the logarithmic function is an inverse of the exponential function, meaning that you still could have mapped the second set of points simply by reflecting over the line y = x, even if the values present did not provide a simple/obvious value for b.(5 votes)

- I've seen a notation such as 10 log (25/5) = 10 log (5). I'm confused about the "10" and the "log" placement. Is this the same as saying log_10 5? Another (very similar) notation is 20 log (0.1/2)=20 log (0.05). Again, why put the 10 and the 20 before the "log"? Thank You!(1 vote)
- Whenever there's no base shown after "log", it means that the base is
**10**. A number before log simply means multiplication. Your example:

is the equivalent of saying:`20 log (0.1/2)`

`20 * log_10 (0.1/2)`

(8 votes)

## Video transcript

Voiceover:The three points plotted below are on the graph of y is
equal to b to the x power. Based only on these three points, plot the three corresponding points that must be on the graph of y is equal to log base b of x by clicking on the graph. I've actually copy and pasted this problem on my little scratch
pad so I can mark it up a little bit. What is this first function? This first function is telling us so x, and this is y is
equal to b to the x power. When x is equal to
zero, y is equal to one. That's this point right over here. When x is equal to one, b to the one power or b to the first power is equal to four. y is equal to four. Another way of thinking of this y or four is equal to b to the first power and actually we can do
[said] must be four. That's this point right over there. This point is telling us
that b to the second power is equal to 16. When x is equal to two, b to the second power, y is equal to 16. Now we want to plot the
three corresponding points on this function. Let me draw another table here. Now it's essentially the inverse function where this is going to be x and we want to calculate y
is equal to log base b of x. What are the possibilities here? What I want to do is think ... Let's take these values because these are
essentially inverse functions log is the inverse of exponents. If we take the points one, four, and 16. What is y going to be here? y is going to be log base b of one. This is saying what power
I need to raise b to to get to one. If we assume that b is non zero and that's a reasonable assumption because b to different
powers are non zero, this is going to be
zero for any non zero b. This is going to be zero
right there, over here. We have the point one comma zero, so it's that point over there. Notice this point
corresponds to this point, we have essentially
swapped the x's and y's. In general when you're taking an inverse you're going to reflect over
the line, y is equal to x and this is clearly
reflection over that line. Now let's look over here, when x is equal to four
what is log base b of four. What is the power I need to raise b to to get to four. We see right over here, b to the first power is equal to four. We already figured that out, when I take b to the first
power is equal to four. This right over here is
going to be equal to one. When x is equal to
four, y is equal to one. Notice once again, it is
a reflection over the line y is equal to x. When x is equal to 16 then y is equal to log base b of 16. The power I need to
raise b to, to get to 16. Well we already know, if we take b squared, we get to 16, so this is equal to two. When x is equal to 16, y is equal to two. Notice we essentially just
swapped the x and y values for each of these points. This is y, this is a reflection over the line y is equal to x. Now, let's actually do that
on the actual interface. The whole reason is to
give you this appreciation that these are inverse
functions of each other. Let's plot the points. That point corresponded to that point, so x zero, y one corresponds to x one, y zero. Here x is one, y is four that corresponds to x four, y one. Here x is two, y is 16 that corresponds to x 16, y is two. We got it right.