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Finding the Inverse of ln(x)

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  • 0:04 Steps to Solving
  • 0:37 Step 1 and 2
  • 0:52 Step 3
  • 1:36 Step 4
  • 2:19 Checking Your Work
  • 5:00 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

How do you find the inverse function of ln(x)? After we go over the steps involved with this process, we'll look at how to check our work in two different ways using rules and graphs.

Steps to Solving

Say we want to find the inverse of the function f(x) = ln(x). In general, we use the notation f -1 (x) to indicate the inverse of the function f(x). To find the inverse of a function, we use the following steps:

  1. Replace f(x) with y.
  2. Interchange x and y.
  3. Solve for y.
  4. Replace y with f -1 (x).

Let's take our function, f(x) = ln(x), through these steps.

Step 1 and 2

First, we want to replace f(x) with y.

y = ln(x)

The second step is to interchange x and y, so we just swap x and y in the equation.

x = ln(y)

Step 3

The third step is to solve for y. To do this, we just need to put x = ln(y) into exponential form. First, we need to recognize that ln(y) is a logarithm with base e, where e is an irrational number with an approximate value of 2.71828.

ln(y) = log e (y).

Now, we just need to use the following rule to put x = ln(y) into exponential form. For all logarithms and exponents:


invlnx1


By the rule, we have that x = ln(y) is equivalent to y = e x. Great! We've solved for y.

Step 4

The last step is to simply replace y with f -1 (x).

f -1 (x) = e x

Let's be a little more organized about this, and display all this work in a nice compact manner.


invlnx2


We see that the inverse of f(x) = ln(x) is f -1 (x) = e x.

Checking Your Work

We're going to look at a couple of ways to check our work. The first makes use of the rule of inverse functions.


invlnx3


This rule states that if we plug f into f -1 or f -1 into f and simplify, we will get x out in both instances. Thus, all we have to do is plug each function into the other and see if we get x back out in both cases.

First, let's plug f into f -1.

f -1 (f(x)) = f -1 (ln(x)) = e ln(x)

We need to simplify e ln(x). In general, log a (b) is the number we need to raise a to in order to get b, so in our case here, we're raising e to the number we need to raise e to in order to get x, it makes sense that we get x out. All together, we have the following.

f -1 (f(x)) = f -1 (ln(x)) = e ln(x) = x

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