From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Logarithms and Exponential Functions

Logarithms and Exponential Functions

Home Algebra II Logarithms and Exponential Functions Topics Linking Exponents and Logarithms

At a Glance - Linking Exponents and Logarithms


You might be wondering how Expo and Log got together in the first place. It couldn't just be Expo's charming good looks, right? In this section, you're going to learn all about how logarithms and exponents are connected—how they're the same and how they're total opposites of one another.

What's a Log, Anyway?

Before we get into the nitty-gritty of how logs work, let's take a gander at how they look. Here's one:

log10x

Weird, right? Why is there a number below the log? What's the log doing to the x? Sometimes you've got to know what's inside to know how something works. Just like in biology class, except as we cut into exponents and logarithms it won't be nearly as squishy or disgusting. We promise.

Example 1

Rewrite this logarithm in exponential form:

y = log(x + 3)


Example 2

Rewrite this exponential equation in logarithmic form:

y = 4(7x)


Example 3

Simplify the following expression:

5log525


Example 4

What is the inverse function of y = 15x + 25?


Example 5

What is the inverse function of y = 10x – 4?


Example 6

Simplify:

3log34


Example 7

Does this function have an inverse?

f(x) = x2 + 3x + 4


Exercise 1

What is the inverse function of y = log 4x ?


Exercise 2

What is the inverse function of y = ln x?


Exercise 3

What is the inverse function of y = 15x + 3?


Exercise 4

What is the inverse function of y = 5x + 1?


Exercise 5

Is y = 4x the inverse of logx 4 = y?


Exercise 6

If f(x) = 2x + 2, what is f(f -1(x))?


Exercise 7

What is the inverse function of y = log4 (3x + 2)?


Exercise 8

What is the inverse function of y = 3x + 1?


Exercise 9

Is f(x) = x2 a one-to-one function?


Exercise 10

Solve for x:

y = 4x – 3


Exercise 11

Solve for x:

y = 10ex


Exercise 12

What is the inverse function of y = 7(x + 10)?


Exercise 13

Is the following function one-to-one?

f(x) = x3 + 4


Exercise 14

How should we restrict the domain of this function to make it one-to-one?

f(x) = 2x2 – 4


People who Shmooped this also Shmooped...

Advertisement