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 2\blueD2 raised to the 4th\greenE4^\text{th} power equals 16\goldD{16}. This is expressed by the exponential equation 24=16\blueD2^\greenE4=\goldD{16}.
Now, suppose someone asked us, "2\blueD2 raised to which power equals 16\goldD{16}?" The answer would be 4\greenE4. This is expressed by the logarithmic equation log2(16)=4\log_\blueD2(\goldD{16})=\greenE4, read as "log base two of sixteen is four".
24=16log2(16)=4\Large \blueD2^\greenE4=\goldD{16}\quad\iff\quad\log_\blueD2(\goldD{16})=\greenE4
Both equations describe the same relationship between the numbers 2\blueD2, 4\greenE4, and 16\goldD{16}, where 2\blueD2 is the base and 4\greenE4 is the exponent.
The difference is that while the exponential form isolates the power, 16\goldD{16}, the logarithmic form isolates the exponent, 4\greenD 4.
Here are more examples of equivalent logarithmic and exponential equations.
Logarithmic formExponential form
log2(8)=3\log_\blueD2(\goldD{8})=\greenD3\iff23=8\blueD2^\greenD3=\goldD8
log3(81)=4\log_\blueD3(\goldD{81})=\greenD4\iff34=81\blueD3^\greenD4=\goldD{81}
log5(25)=2\log_\blueD5(\goldD{25})=\greenD2\iff52=25\blueD5^\greenD2=\goldD{25}

Definition of a logarithm

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

A helpful note

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.

Check your understanding

In the following problems, you will convert between exponential and logarithmic forms of equations.
1) Which of the following is equivalent to 25=322^5=32?
Choose 1 answer:
Choose 1 answer:

We know that bc=a \blueD b^\greenD c=\goldD a is equivalent to logb(a)=c\log_\blueD b(\goldD a)=\greenD c.
Using this equivalence, we can rewrite 25=32\blueD 2^\greenD 5=\goldD{32} as log2(32)=5\log_\blueD 2(\goldD{32})=\greenD{5}.
The answer is log2(32)=5\log_2(32)=5.
2) Which of the following is equivalent to 53=1255^3=125?
Choose 1 answer:
Choose 1 answer:

We know that bc=a \blueD b^\greenD c=\goldD a is equivalent to logb(a)=c\log_\blueD b(\goldD a)=\greenD c.
Using this equivalence, we can rewrite 53=125\blueD 5^\greenD 3=\goldD{125} as log5(125)=3\log_\blueD 5(\goldD{125})=\greenD{3}.
The answer is log5(125)=3\log_5(125)=3.
3) Write log2(64)=6\log_2(64)=6 in exponential form.

We know that logb(a)=c\log_\blueD b(\goldD a)=\greenD c is equivalent to bc=a \blueD b^\greenD c=\goldD a.
Using this equivalence, we can rewrite log2(64)=6\log_\blueD 2(\goldD{64})=\greenD{6} as 26=64\blueD 2^\greenD 6=\goldD{64}.
The answer is 26=642^6=64.
4) Write log4(16)=2\log_4(16)=2 in exponential form.

We know that logb(a)=c\log_\blueD b(\goldD a)=\greenD c is equivalent to bc=a \blueD b^\greenD c=\goldD a.
Using this equivalence, we can rewrite log4(16)=2\log_\blueD 4(\goldD{16})=\greenD{2} as 42=16\blueD 4^\greenD 2=\goldD{16}.
The answer is 42=164^2=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 log4(64)\log_4(64).
Let's start by setting that expression equal to xx.
log4(64)=x\log_4(64)=x
Writing this as an exponential equation gives us the following:
4x=644^x=64
44 to what power is 6464? Well, 43=64\blueD4^\greenD 3=\goldD{64} and so log4(64)=3\log_\blueD4(\goldD{64})=\greenD3.
As you become more practiced, you may find yourself condensing a few of these steps and evaluating log4(64)\log_4(64) just by asking, "44 to what power is 6464?"

Check your understanding

Remember, when evaluating logb(a)\log_\blueD{b}(\goldD{a}), you can ask: "b\blueD b to what power is a\goldD a?"
5) log6(36)=\log_6(36) =
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Let log6(36)=x\log_6(36)=x. This is equivalent to 6x=366^x=36.
66 to what power is 3636?
62=366^\tealD2=36, so log6(36)=2\log_6(36)=\tealD2.
6) log3(27)=\log_3(27)=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Let log3(27)=x\log_3(27)=x. This is equivalent to 3x=273^x=27.
33 to what power is 2727?
33=273^\tealD 3=27, so log3(27)=3\log_3(27)=\tealD 3.
7) log4(4)=\log_4(4)=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Let log4(4)=x\log_4(4)=x. This is equivalent to 4x=44^x=4.
44 to what power is 44?
41=44^\tealD 1=4, so log4(4)=1\log_4(4)=\tealD 1.
8) log5(1)=\log_5(1)=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Let log5(1)=x\log_5(1)=x. This is equivalent to 5x=15^x=1.
55 to what power is equal to 11?
50=15^0=1, so log5(1)=0\log_5(1)=0.
In fact, logb(1)=0\log_b(1)=0 for all possible valid bases. This is because any number raised to the zero power is 11.

Challenge Problem

9*) log3(19)=\log_3\left(\dfrac19\right)=
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}

Let log3(19)=x\log_3\left(\dfrac19\right)=x. This is equivalent to 3x=193^x=\dfrac19.
33 to what power is 19\dfrac19?
32=193^\tealD {-2}=\dfrac19, so log3(19)=2\log_3\left(\dfrac19\right)=\tealD {-2}.

Restrictions on the variables

logb(a)\log_b(a) is defined when the base bb is positive—and not equal to 11—and the argument aa is positive. These restrictions are a result of the connection between logarithms and exponents.
RestrictionReasoning
b>0b>0In an exponential function, the base bb is always defined to be positive.
a>0a>0logb(a)=c\log_b(a)=c means that bc=ab^c=a. Because a positive number raised to any power is positive, meaning bc>0b^c>0, it follows that a>0a>0.
b1b\neq1Suppose, for a moment, that bb could be 11. Now consider the equation log1(3)=x\log_1(3)=x. The equivalent exponential form would be 1x=31^x=3. But this can never be true since 11 to any power is always 11. So, it follows that b1b\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 1010 ("base-1010 logarithm").
When writing these logarithms mathematically, we omit the base. It is understood to be 1010.
log10(x)=log(x)\log_{10}{(x)}=\log(x)

The natural logarithm

The natural logarithm is a logarithm whose base is the number ee ("base-ee logarithm").
ee 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 ee as you would any other number.
Instead of writing the base as ee, we indicate the logarithm with ln\ln.
loge(x)=ln(x)\log_e(x)=\ln(x)
This table summarizes what we need to know about these two special logarithms:
NameBaseRegular notationSpecial notation
Common logarithm1010log10(x)\log_{10}(x)log(x)\log(x)
Natural logarithmeeloge(x)\log_e(x)ln(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)\log(100).

Solution 1

By definition, log(100)=log10(100)\log(100)=\log_{10}(100).
1010 to what power is 100100?
102=10010^\tealD 2=100, so log(100)=2\log(100)=\tealD 2.

Example 2

Evaluate ln(e3)\ln(e^3).

Solution 2

By definition, ln(e3)=loge(e3)\ln(e^3)=\log_{e}(e^3).
ee to what power is e3e^3?
e3=e3e^\tealD3=e^3, so ln(e3)=3\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 2x=52^x=5 can be given as a logarithm, x=log2(5)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.
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