# Writing the Inverse of Logarithmic Functions

Writing the Inverse of Logarithmic Functions

Amy has a master’s degree in secondary education and has taught math at a public charter high school.

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Watch this video lesson to learn how inverses are related to the original function. Also learn a method to find the inverse of logarithmic functions that you can easily use.

The Logarithmic Function

The logarithmic function is f(x) = log subb(x), where b is a number greater than 0 but not 1. The b is also called the base of the logarithmic function. Also, when referring to the logarithm, the shortened version, log, is often used. You’ll see me use the shortened version ‘log’ in this video lesson as well as the whole word logarithm.

The log function.

inverse log functions

When the base b of the logarithm equals 10, we normally don’t write it out. So our function with a base of 10 is written simply as f(x) = log(x). We don’t write the 10 because 10 is the standard base, and when no base is written, it is understood that it is the standard 10. You can remember that 10 is the standard base by looking at the number of fingers and toes that most everyone has. People generally have a standard of 10 fingers and 10 toes.

There is another base that we don’t write, and that is the number e, which is approximately 2.71828. When e is the base, we call the log the natural log and we write out the function as f(x) = ln(x).

For all other bases, we write it as a subscript immediately following the word ‘log’ in functions.

The Inverse Function

The inverse of a function is the reverse of the function. The notation for the inverse is f^-1(y). Using this notation, if f(a) = c, then f^-1(c) = a. Both of these statements say the same thing, and we can use either formula to write this same relationship. Plugging in numbers, if f(1) = 2, then f^-1(2) = 1. You can think of this as the function going one way and the inverse function going in the opposite direction. We can combine all this information into one formula: f(f^-1(x)) = x.

Inverse functions.

inverse log functions

Some functions in math have a known inverse function. The log function is one of these functions. We know that the inverse of a log function is an exponential. So, we know that the inverse of f(x) = log subb(x) is f^-1(y) = b^y. If the base is e and we are dealing with the natural log, then the inverse of f(x) = ln(x) is f^-1(y) = e^y.

Rewriting these relationships substituting f(x) with y and f^-1(y) with x, we see that the inverse of y = log subb(x) is x = b^y and the inverse of y = ln(x) is x = e^y. Both the inverse and the original function state the same relationship, so we can use either formula. It is like the formulas for converting temperature from Fahrenheit to Celsius and back again. We can use either of these formulas to state the same relationship.

Function and inverse function equivalents.

inverse log functions

We are going to use this information to help us figure out the inverse of other functions that involve logs.

Setting Up the Problem

Let’s see how this is done.

Let’s say we want to find the inverse of the function f(x) = log sub3(x + 2) – 4.

The original function.

inverse log functions

To set up our problem, we will use what we know of inverses, that the function of an inverse is simply x. If we plug in our inverse for our x, then our function will equal x itself. Let’s do that.

Setting up the problem.

inverse log functions

Notice that I have plugged in the notation for our inverse function wherever I have an x.

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