# Intro to Logarithms

CCSS Math: HSF.BF.B.5
Learn what logarithms are and how to evaluate them.

### What you should be familiar with before taking this lesson

You should be familiar with exponents, preferably including negative exponents.

### What you will learn in this lesson

You will learn what logarithms are, and evaluate some basic logarithms. This will prepare you for future work with logarithm expressions and functions.

## What is a logarithm?

Logarithms are another way of thinking about exponents.
For example, we know that $\blueD2$ raised to the $\greenE4^\text{th}$ power equals $\goldD{16}$. This is expressed by the exponential equation $\blueD2^\greenE4=\goldD{16}$.
Now, suppose someone asked us, "$\blueD2$ raised to which power equals $\goldD{16}$?" The answer would be $\greenE4$. This is expressed by the logarithmic equation $\log_\blueD2(\goldD{16})=\greenE4$, read as "log base two of sixteen is four".
$\Large \blueD2^\greenE4=\goldD{16}\quad\iff\quad\log_\blueD2(\goldD{16})=\greenE4$
Both equations describe the same relationship between the numbers $\blueD2$, $\greenE4$, and $\goldD{16}$, where $\blueD2$ is the base and $\greenE4$ is the exponent.
The difference is that while the exponential form isolates the power, $\goldD{16}$, the logarithmic form isolates the exponent, $\greenD 4$.
Here are more examples of equivalent logarithmic and exponential equations.
Logarithmic formExponential form
$\log_\blueD2(\goldD{8})=\greenD3$$\iff$$\blueD2^\greenD3=\goldD8$
$\log_\blueD3(\goldD{81})=\greenD4$$\iff$$\blueD3^\greenD4=\goldD{81}$
$\log_\blueD5(\goldD{25})=\greenD2$$\iff$$\blueD5^\greenD2=\goldD{25}$

## Definition of a logarithm

Generalizing the examples above leads us to the formal definition of a logarithm.
$\Large\log_\blueD b(\goldD a)=\greenD c\quad \iff\quad \blueD b^\greenD c=\goldD a$
Both equations describe the same relationship between $\goldD a$, $\blueD b$, and $\greenE c$:
• $\blueD b$ is the $\blueD{\text{base}}$,
• $\greenE c$ is the $\greenE{\text{exponent}}$, and
• $\goldD a$ is called the $\goldD{\text{argument}}$.

When rewriting an exponential equation in log form or a log equation in exponential form, it is helpful to remember that the base of the logarithm is the same as the base of the exponent.

In the following problems, you will convert between exponential and logarithmic forms of equations.
1) Which of the following is equivalent to $2^5=32$?

We know that $\blueD b^\greenD c=\goldD a$ is equivalent to $\log_\blueD b(\goldD a)=\greenD c$.
Using this equivalence, we can rewrite $\blueD 2^\greenD 5=\goldD{32}$ as $\log_\blueD 2(\goldD{32})=\greenD{5}$.
The answer is $\log_2(32)=5$.
2) Which of the following is equivalent to $5^3=125$?

We know that $\blueD b^\greenD c=\goldD a$ is equivalent to $\log_\blueD b(\goldD a)=\greenD c$.
Using this equivalence, we can rewrite $\blueD 5^\greenD 3=\goldD{125}$ as $\log_\blueD 5(\goldD{125})=\greenD{3}$.
The answer is $\log_5(125)=3$.
3) Write $\log_2(64)=6$ in exponential form.

We know that $\log_\blueD b(\goldD a)=\greenD c$ is equivalent to $\blueD b^\greenD c=\goldD a$.
Using this equivalence, we can rewrite $\log_\blueD 2(\goldD{64})=\greenD{6}$ as $\blueD 2^\greenD 6=\goldD{64}$.
The answer is $2^6=64$.
4) Write $\log_4(16)=2$ in exponential form.

We know that $\log_\blueD b(\goldD a)=\greenD c$ is equivalent to $\blueD b^\greenD c=\goldD a$.
Using this equivalence, we can rewrite $\log_\blueD 4(\goldD{16})=\greenD{2}$ as $\blueD 4^\greenD 2=\goldD{16}$.
The answer is $4^2=16$.

## Evaluating logarithms

Great! Now that we understand the relationship between exponents and logarithms, let's see if we can evaluate logarithms.
For example, let's evaluate $\log_4(64)$.
Let's start by setting that expression equal to $x$.
$\log_4(64)=x$
Writing this as an exponential equation gives us the following:
$4^x=64$
$4$ to what power is $64$? Well, $\blueD4^\greenD 3=\goldD{64}$ and so $\log_\blueD4(\goldD{64})=\greenD3$.
As you become more practiced, you may find yourself condensing a few of these steps and evaluating $\log_4(64)$ just by asking, "$4$ to what power is $64$?"

Remember, when evaluating $\log_\blueD{b}(\goldD{a})$, you can ask: "$\blueD b$ to what power is $\goldD a$?"
5) $\log_6(36) =$

Let $\log_6(36)=x$. This is equivalent to $6^x=36$.
$6$ to what power is $36$?
$6^\tealD2=36$, so $\log_6(36)=\tealD2$.
6) $\log_3(27)=$

Let $\log_3(27)=x$. This is equivalent to $3^x=27$.
$3$ to what power is $27$?
$3^\tealD 3=27$, so $\log_3(27)=\tealD 3$.
7) $\log_4(4)=$

Let $\log_4(4)=x$. This is equivalent to $4^x=4$.
$4$ to what power is $4$?
$4^\tealD 1=4$, so $\log_4(4)=\tealD 1$.
8) $\log_5(1)=$

Let $\log_5(1)=x$. This is equivalent to $5^x=1$.
$5$ to what power is equal to $1$?
$5^0=1$, so $\log_5(1)=0$.
In fact, $\log_b(1)=0$ for all possible valid bases. This is because any number raised to the zero power is $1$.

### Challenge Problem

9*) $\log_3\left(\dfrac19\right)=$

Let $\log_3\left(\dfrac19\right)=x$. This is equivalent to $3^x=\dfrac19$.
$3$ to what power is $\dfrac19$?
$3^\tealD {-2}=\dfrac19$, so $\log_3\left(\dfrac19\right)=\tealD {-2}$.

## Restrictions on the variables

$\log_b(a)$ is defined when the base $b$ is positive—and not equal to $1$—and the argument $a$ is positive. These restrictions are a result of the connection between logarithms and exponents.
RestrictionReasoning
$b>0$In an exponential function, the base $b$ is always defined to be positive.
$a>0$$\log_b(a)=c$ means that $b^c=a$. Because a positive number raised to any power is positive, meaning $b^c>0$, it follows that $a>0$.
$b\neq1$Suppose, for a moment, that $b$ could be $1$. Now consider the equation $\log_1(3)=x$. The equivalent exponential form would be $1^x=3$. But this can never be true since $1$ to any power is always $1$. So, it follows that $b\neq1$.

## Special logarithms

While the base of a logarithm can have many different values, there are two bases that are used more often than others.
Specifically, most calculators have buttons for only these two types of logarithms. Let's check them out.

### The common logarithm

The common logarithm is a logarithm whose base is $10$ ("base-$10$ logarithm").
When writing these logarithms mathematically, we omit the base. It is understood to be $10$.
$\log_{10}{(x)}=\log(x)$

### The natural logarithm

The natural logarithm is a logarithm whose base is the number $e$ ("base-$e$ logarithm").
$e$ is a mathematical constant. It is an irrational number that is approximately equal to 2.718. It appears in many contexts that involve limits, which you will likely learn about as you study calculus. For now, just treat $e$ as you would any other number.
Instead of writing the base as $e$, we indicate the logarithm with $\ln$.
$\log_e(x)=\ln(x)$
This table summarizes what we need to know about these two special logarithms:
NameBaseRegular notationSpecial notation
Common logarithm$10$$\log_{10}(x)$$\log(x)$
Natural logarithm$e$$\log_e(x)$$\ln(x)$

While the notation is different, the idea behind evaluating the logarithm is exactly the same!
Sure! Here are two examples with solutions.

#### Example 1

Evaluate $\log(100)$.

#### Solution 1

By definition, $\log(100)=\log_{10}(100)$.
$10$ to what power is $100$?
$10^\tealD 2=100$, so $\log(100)=\tealD 2$.

#### Example 2

Evaluate $\ln(e^3)$.

#### Solution 2

By definition, $\ln(e^3)=\log_{e}(e^3)$.
$e$ to what power is $e^3$?
$e^\tealD3=e^3$, so $\ln(e^3)=\tealD3$.

## Why are we studying logarithms?

As you just learned, logarithms reverse exponents. For this reason, they are very helpful for solving exponential equations.
For example the result for $2^x=5$ can be given as a logarithm, $x=\log_2(5)$. You will learn how to evaluate this logarithmic expression over the following lessons.
Logarithmic expressions and functions also turn out to be very interesting by themselves, and are actually very common in the world around us. For example, many physical phenomena are measured with logarithmic scales.

## What's next?

Learn about the properties of logarithms that help us rewrite logarithmic expressions, and about the change of base rule that allows us to evaluate any logarithm we want using the calculator.