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# Intro to Logarithms

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 start color #11accd, 2, end color #11accd raised to the start color #0d923f, 4, end color #0d923f, start superscript, start text, t, h, end text, end superscript power equals start color #e07d10, 16, end color #e07d10. This is expressed by the exponential equation start color #11accd, 2, end color #11accd, start superscript, start color #0d923f, 4, end color #0d923f, end superscript, equals, start color #e07d10, 16, end color #e07d10.
Now, suppose someone asked us, "start color #11accd, 2, end color #11accd raised to which power equals start color #e07d10, 16, end color #e07d10?" The answer would be start color #0d923f, 4, end color #0d923f. This is expressed by the logarithmic equation log, start base, start color #11accd, 2, end color #11accd, end base, left parenthesis, start color #e07d10, 16, end color #e07d10, right parenthesis, equals, start color #0d923f, 4, end color #0d923f, read as "log base two of sixteen is four".
start color #11accd, 2, end color #11accd, start superscript, start color #0d923f, 4, end color #0d923f, end superscript, equals, start color #e07d10, 16, end color #e07d10, \Longleftrightarrow, log, start base, start color #11accd, 2, end color #11accd, end base, left parenthesis, start color #e07d10, 16, end color #e07d10, right parenthesis, equals, start color #0d923f, 4, end color #0d923f
Both equations describe the same relationship between the numbers start color #11accd, 2, end color #11accd, start color #0d923f, 4, end color #0d923f, and start color #e07d10, 16, end color #e07d10, where start color #11accd, 2, end color #11accd is the base and start color #0d923f, 4, end color #0d923f is the exponent.
The difference is that while the exponential form isolates the power, start color #e07d10, 16, end color #e07d10, the logarithmic form isolates the exponent, start color #1fab54, 4, end color #1fab54.
Here are more examples of equivalent logarithmic and exponential equations.
Logarithmic formExponential form
log, start base, start color #11accd, 2, end color #11accd, end base, left parenthesis, start color #e07d10, 8, end color #e07d10, right parenthesis, equals, start color #1fab54, 3, end color #1fab54\Longleftrightarrowstart color #11accd, 2, end color #11accd, start superscript, start color #1fab54, 3, end color #1fab54, end superscript, equals, start color #e07d10, 8, end color #e07d10
log, start base, start color #11accd, 3, end color #11accd, end base, left parenthesis, start color #e07d10, 81, end color #e07d10, right parenthesis, equals, start color #1fab54, 4, end color #1fab54\Longleftrightarrowstart color #11accd, 3, end color #11accd, start superscript, start color #1fab54, 4, end color #1fab54, end superscript, equals, start color #e07d10, 81, end color #e07d10
log, start base, start color #11accd, 5, end color #11accd, end base, left parenthesis, start color #e07d10, 25, end color #e07d10, right parenthesis, equals, start color #1fab54, 2, end color #1fab54\Longleftrightarrowstart color #11accd, 5, end color #11accd, start superscript, start color #1fab54, 2, end color #1fab54, end superscript, equals, start color #e07d10, 25, end color #e07d10

## Definition of a logarithm

Generalizing the examples above leads us to the formal definition of a logarithm.
log, start base, start color #11accd, b, end color #11accd, end base, left parenthesis, start color #e07d10, a, end color #e07d10, right parenthesis, equals, start color #1fab54, c, end color #1fab54, \Longleftrightarrow, start color #11accd, b, end color #11accd, start superscript, start color #1fab54, c, end color #1fab54, end superscript, equals, start color #e07d10, a, end color #e07d10
Both equations describe the same relationship between start color #e07d10, a, end color #e07d10, start color #11accd, b, end color #11accd, and start color #0d923f, c, end color #0d923f:
• start color #11accd, b, end color #11accd is the start color #11accd, start text, b, a, s, e, end text, end color #11accd,
• start color #0d923f, c, end color #0d923f is the start color #0d923f, start text, e, x, p, o, n, e, n, t, end text, end color #0d923f, and
• start color #e07d10, a, end color #e07d10 is called the start color #e07d10, start text, a, r, g, u, m, e, n, t, end text, end color #e07d10.

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.
Problem 1
Which of the following is equivalent to 2, start superscript, 5, end superscript, equals, 32?

Problem 2
Which of the following is equivalent to 5, cubed, equals, 125?

Problem 3
Write log, start base, 2, end base, left parenthesis, 64, right parenthesis, equals, 6 in exponential form.

Problem 4
4) Write log, start base, 4, end base, left parenthesis, 16, right parenthesis, equals, 2 in exponential form.

## 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, start base, 4, end base, left parenthesis, 64, right parenthesis.
Let's start by setting that expression equal to x.
log, start base, 4, end base, left parenthesis, 64, right parenthesis, equals, x
Writing this as an exponential equation gives us the following:
4, start superscript, x, end superscript, equals, 64
4 to what power is 64? Well, start color #11accd, 4, end color #11accd, start superscript, start color #1fab54, 3, end color #1fab54, end superscript, equals, start color #e07d10, 64, end color #e07d10 and so log, start base, start color #11accd, 4, end color #11accd, end base, left parenthesis, start color #e07d10, 64, end color #e07d10, right parenthesis, equals, start color #1fab54, 3, end color #1fab54.
As you become more practiced, you may find yourself condensing a few of these steps and evaluating log, start base, 4, end base, left parenthesis, 64, right parenthesis just by asking, "4 to what power is 64?"

Remember, when evaluating log, start base, start color #11accd, b, end color #11accd, end base, left parenthesis, start color #e07d10, a, end color #e07d10, right parenthesis, you can ask: "start color #11accd, b, end color #11accd to what power is start color #e07d10, a, end color #e07d10?"
Problem 5
log, start base, 6, end base, left parenthesis, 36, right parenthesis, equals

Problem 6
log, start base, 3, end base, left parenthesis, 27, right parenthesis, equals

Problem 7
log, start base, 4, end base, left parenthesis, 4, right parenthesis, equals

Problem 8
log, start base, 5, end base, left parenthesis, 1, right parenthesis, equals

Challenge problem
log, start base, 3, end base, left parenthesis, start fraction, 1, divided by, 9, end fraction, right parenthesis, equals

## Restrictions on the variables

log, start base, b, end base, left parenthesis, a, right parenthesis 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, is greater than, 0In an exponential function, the base b is always defined to be positive.
a, is greater than, 0log, start base, b, end base, left parenthesis, a, right parenthesis, equals, c means that b, start superscript, c, end superscript, equals, a. Because a positive number raised to any power is positive, meaning b, start superscript, c, end superscript, is greater than, 0, it follows that a, is greater than, 0.
b, does not equal, 1Suppose, for a moment, that b could be 1. Now consider the equation log, start base, 1, end base, left parenthesis, 3, right parenthesis, equals, x. The equivalent exponential form would be 1, start superscript, x, end superscript, equals, 3. But this can never be true since 1 to any power is always 1. So, it follows that b, does not equal, 1.

## 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, start base, 10, end base, left parenthesis, x, right parenthesis, equals, log, left parenthesis, x, right parenthesis

### The natural logarithm

The natural logarithm is a logarithm whose base is the number e ("base-e logarithm").
Instead of writing the base as e, we indicate the logarithm with natural log.
log, start base, e, end base, left parenthesis, x, right parenthesis, equals, natural log, left parenthesis, x, right parenthesis
This table summarizes what we need to know about these two special logarithms:
NameBaseRegular notationSpecial notation
Common logarithm10log, start base, 10, end base, left parenthesis, x, right parenthesislog, left parenthesis, x, right parenthesis
Natural logarithmelog, start base, e, end base, left parenthesis, x, right parenthesisnatural log, left parenthesis, x, right parenthesis
While the notation is different, the idea behind evaluating the logarithm is exactly the same!

## 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, start superscript, x, end superscript, equals, 5 can be given as a logarithm, x, equals, log, start base, 2, end base, left parenthesis, 5, right parenthesis. 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.