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

Definition of a logarithm

Generalizing the examples above leads us to the formal definition of a logarithm.
Both equations describe the same relationship between start color goldD, a, end color goldD, start color blueD, b, end color blueD, and start color greenE, c, end color greenE:
• start color blueD, b, end color blueD is the start color blueD, b, a, s, e, end color blueD,
• start color greenE, c, end color greenE is the start color greenE, e, x, p, o, n, e, n, t, end color greenE, and
• start color goldD, a, end color goldD is the start color goldD, p, o, w, e, r, end color goldD.
In logarithm form, this is also called the start color goldD, a, r, g, u, m, e, n, t, end color goldD.

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, start superscript, 5, end superscript, equals, 32?

We know that start color blueD, b, end color blueD, start superscript, start color greenD, c, end color greenD, end superscript, equals, start color goldD, a, end color goldD is equivalent to log, start subscript, start color blueD, b, end color blueD, end subscript, left parenthesis, start color goldD, a, end color goldD, right parenthesis, equals, start color greenD, c, end color greenD.
Using this equivalence, we can rewrite start color blueD, 2, end color blueD, start superscript, start color greenD, 5, end color greenD, end superscript, equals, start color goldD, 32, end color goldD as log, start subscript, start color blueD, 2, end color blueD, end subscript, left parenthesis, start color goldD, 32, end color goldD, right parenthesis, equals, start color greenD, 5, end color greenD.
The answer is log, start subscript, 2, end subscript, left parenthesis, 32, right parenthesis, equals, 5.
2) Which of the following is equivalent to 5, start superscript, 3, end superscript, equals, 125?

We know that start color blueD, b, end color blueD, start superscript, start color greenD, c, end color greenD, end superscript, equals, start color goldD, a, end color goldD is equivalent to log, start subscript, start color blueD, b, end color blueD, end subscript, left parenthesis, start color goldD, a, end color goldD, right parenthesis, equals, start color greenD, c, end color greenD.
Using this equivalence, we can rewrite start color blueD, 5, end color blueD, start superscript, start color greenD, 3, end color greenD, end superscript, equals, start color goldD, 125, end color goldD as log, start subscript, start color blueD, 5, end color blueD, end subscript, left parenthesis, start color goldD, 125, end color goldD, right parenthesis, equals, start color greenD, 3, end color greenD.
The answer is log, start subscript, 5, end subscript, left parenthesis, 125, right parenthesis, equals, 3.
3) Write log, start subscript, 2, end subscript, left parenthesis, 64, right parenthesis, equals, 6 in exponential form.

We know that log, start subscript, start color blueD, b, end color blueD, end subscript, left parenthesis, start color goldD, a, end color goldD, right parenthesis, equals, start color greenD, c, end color greenD is equivalent to start color blueD, b, end color blueD, start superscript, start color greenD, c, end color greenD, end superscript, equals, start color goldD, a, end color goldD.
Using this equivalence, we can rewrite log, start subscript, start color blueD, 2, end color blueD, end subscript, left parenthesis, start color goldD, 64, end color goldD, right parenthesis, equals, start color greenD, 6, end color greenD as start color blueD, 2, end color blueD, start superscript, start color greenD, 6, end color greenD, end superscript, equals, start color goldD, 64, end color goldD.
The answer is 2, start superscript, 6, end superscript, equals, 64.
4) Write log, start subscript, 4, end subscript, left parenthesis, 16, right parenthesis, equals, 2 in exponential form.

We know that log, start subscript, start color blueD, b, end color blueD, end subscript, left parenthesis, start color goldD, a, end color goldD, right parenthesis, equals, start color greenD, c, end color greenD is equivalent to start color blueD, b, end color blueD, start superscript, start color greenD, c, end color greenD, end superscript, equals, start color goldD, a, end color goldD.
Using this equivalence, we can rewrite log, start subscript, start color blueD, 4, end color blueD, end subscript, left parenthesis, start color goldD, 16, end color goldD, right parenthesis, equals, start color greenD, 2, end color greenD as start color blueD, 4, end color blueD, start superscript, start color greenD, 2, end color greenD, end superscript, equals, start color goldD, 16, end color goldD.
The answer is 4, start superscript, 2, end superscript, equals, 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, start subscript, 4, end subscript, left parenthesis, 64, right parenthesis.
Let's start by setting that expression equal to x.
log, start subscript, 4, end subscript, 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 blueD, 4, end color blueD, start superscript, start color greenD, 3, end color greenD, end superscript, equals, start color goldD, 64, end color goldD and so log, start subscript, start color blueD, 4, end color blueD, end subscript, left parenthesis, start color goldD, 64, end color goldD, right parenthesis, equals, start color greenD, 3, end color greenD.
As you become more practiced, you may find yourself condensing a few of these steps and evaluating log, start subscript, 4, end subscript, left parenthesis, 64, right parenthesis just by asking, "4 to what power is 64?"

Remember, when evaluating log, start subscript, start color blueD, b, end color blueD, end subscript, left parenthesis, start color goldD, a, end color goldD, right parenthesis, you can ask: "start color blueD, b, end color blueD to what power is start color goldD, a, end color goldD?"
5) log, start subscript, 6, end subscript, left parenthesis, 36, right parenthesis, equals

Let log, start subscript, 6, end subscript, left parenthesis, 36, right parenthesis, equals, x. This is equivalent to 6, start superscript, x, end superscript, equals, 36.
6 to what power is 36?
6, start superscript, start color tealD, 2, end color tealD, end superscript, equals, 36, so log, start subscript, 6, end subscript, left parenthesis, 36, right parenthesis, equals, start color tealD, 2, end color tealD.
6) log, start subscript, 3, end subscript, left parenthesis, 27, right parenthesis, equals

Let log, start subscript, 3, end subscript, left parenthesis, 27, right parenthesis, equals, x. This is equivalent to 3, start superscript, x, end superscript, equals, 27.
3 to what power is 27?
3, start superscript, start color tealD, 3, end color tealD, end superscript, equals, 27, so log, start subscript, 3, end subscript, left parenthesis, 27, right parenthesis, equals, start color tealD, 3, end color tealD.
7) log, start subscript, 4, end subscript, left parenthesis, 4, right parenthesis, equals

Let log, start subscript, 4, end subscript, left parenthesis, 4, right parenthesis, equals, x. This is equivalent to 4, start superscript, x, end superscript, equals, 4.
4 to what power is 4?
4, start superscript, start color tealD, 1, end color tealD, end superscript, equals, 4, so log, start subscript, 4, end subscript, left parenthesis, 4, right parenthesis, equals, start color tealD, 1, end color tealD.
8) log, start subscript, 5, end subscript, left parenthesis, 1, right parenthesis, equals

Let log, start subscript, 5, end subscript, left parenthesis, 1, right parenthesis, equals, x. This is equivalent to 5, start superscript, x, end superscript, equals, 1.
5 to what power is equal to 1?
5, start superscript, 0, end superscript, equals, 1, so log, start subscript, 5, end subscript, left parenthesis, 1, right parenthesis, equals, 0.
In fact, log, start subscript, b, end subscript, left parenthesis, 1, right parenthesis, equals, 0 for all possible valid bases. This is because any number raised to the zero power is 1.

Challenge Problem

9*) log, start subscript, 3, end subscript, left parenthesis, start fraction, 1, divided by, 9, end fraction, right parenthesis, equals

Let log, start subscript, 3, end subscript, left parenthesis, start fraction, 1, divided by, 9, end fraction, right parenthesis, equals, x. This is equivalent to 3, start superscript, x, end superscript, equals, start fraction, 1, divided by, 9, end fraction.
3 to what power is start fraction, 1, divided by, 9, end fraction?
3, start superscript, start color tealD, minus, 2, end color tealD, end superscript, equals, start fraction, 1, divided by, 9, end fraction, so log, start subscript, 3, end subscript, left parenthesis, start fraction, 1, divided by, 9, end fraction, right parenthesis, equals, start color tealD, minus, 2, end color tealD.

Restrictions on the variables

log, start subscript, b, end subscript, 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 subscript, b, end subscript, 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 subscript, 1, end subscript, 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 subscript, 10, end subscript, 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").
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 natural log.
log, start subscript, e, end subscript, 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 subscript, 10, end subscript, left parenthesis, x, right parenthesislog, left parenthesis, x, right parenthesis
Natural logarithmelog, start subscript, e, end subscript, 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!
Sure! Here are two examples with solutions.

Example 1

Evaluate log, left parenthesis, 100, right parenthesis.

Solution 1

By definition, log, left parenthesis, 100, right parenthesis, equals, log, start subscript, 10, end subscript, left parenthesis, 100, right parenthesis.
10 to what power is 100?
10, start superscript, start color tealD, 2, end color tealD, end superscript, equals, 100, so log, left parenthesis, 100, right parenthesis, equals, start color tealD, 2, end color tealD.

Example 2

Evaluate natural log, left parenthesis, e, start superscript, 3, end superscript, right parenthesis.

Solution 2

By definition, natural log, left parenthesis, e, start superscript, 3, end superscript, right parenthesis, equals, log, start subscript, e, end subscript, left parenthesis, e, start superscript, 3, end superscript, right parenthesis.
e to what power is e, start superscript, 3, end superscript?
e, start superscript, start color tealD, 3, end color tealD, end superscript, equals, e, start superscript, 3, end superscript, so natural log, left parenthesis, e, start superscript, 3, end superscript, right parenthesis, equals, start color tealD, 3, end color tealD.

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 subscript, 2, end subscript, 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.