Logarithmic And Exponential Functions - Introduction

Exponential and logarithmic equations might seem daunting at first glance. Let’s break down exactly what they are and learn how to solve them!

What Are Exponents and Logarithms?

An exponent is used to show that you’re multiplying something by itself a certain number of times. So,

\({2}^{3}\)

is a quicker way to write

\(2\times 2\times 2\)

— and both give the same answer:

\(8\)

.
Logarithms are the opposite of exponents. Consider this: We know that

\({2}^{3}=8\)

, but let's represent this equation in another way. Using logarithms, we can re-write this equation as

\(\log_{2}{8}=3\)

. The subscript

\(2\)

here is the base of the logarithm. This is similar to asking, "

\(2\)

raised to which power equals

\(8\)

?" The answer would be

\(3\)

.

Useful Logarithm Rules

There are two logarithm rules often used to help solve equations: The natural log rule and the common logarithm rule.
The Natural Log Rule states that if

\({e}^{y}=x\)

, then

\(\ln{x}=y\)

, where

\(e\)

is a transcendental number much like

\(\pi\)

, and is 2.718281828459 approximately.
The Common Logarithm Rule states that if

\({b}^{a}=x\)

, then

\(\log_{b}{x}=a\)

.

Exponent and Logarithm Examples

Let’s look at a couple equations and see these rules in action.
1. Given the equation

\(\ln{x}=3\)

, solve for

\(x\)

.
Since we know that

\(\ln{x}=y\)

is equivalent to

\({e}^{y}=x\)

, we can apply the transformation to this equation and get

\(x={e}^{3}\)

.
2. Given the equation

\({2}^{x}=10\)

, solve for

\(x\)

.
First, we use the common logarithm rule to yield

\(x=\log_{2}{10}\)

. Then, we can use what is called the “change of base rule”, which states that

\(\log_{b}{x}=\frac{\log_{a}{x}}{\log_{a}{b}}\)

. This gives

\(x=\frac{\log{10}}{\log{2}}\)

. The Rule of

\(10\)

then allows us to convert

\(\log_{10}{}\)

into

\(1\)

to give the final answer of

\(x=\frac{1}{\log{2}}\)

.

What's Next

Getting comfortable with exponential and logarithmic functions takes patience and practice. Start with our exponential and logarithmic equation practice problems at the top of this page — see how long it takes you to solve them. Need more help? Sign up for Cymath Plus today.
Keep in mind that Cymath can help you tackle everything from Pre-Algebra to Calculus. Make sure you also give other practice problems a try, and best of luck in your studies!