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# 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?

Solve for x:

y = 4x – 3

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