Evaluating Logarithms

Learning Outcomes

Evaluate logarithms with and without a calculator.
Evaluate logarithms with base 10 and base e.

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider ${\mathrm{log}}_{2}8$ . We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know ${2}^{3}=8$ , it follows that ${\mathrm{log}}_{2}8=3$ .

Now consider solving ${\mathrm{log}}_{7}49$ and ${\mathrm{log}}_{3}27$ mentally.

We ask, “To what exponent must 7 be raised in order to get 49?” We know ${7}^{2}=49$ . Therefore, ${\mathrm{log}}_{7}49=2$ .
We ask, “To what exponent must 3 be raised in order to get 27?” We know ${3}^{3}=27$ . Therefore, ${\mathrm{log}}_{3}27=3$ .

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate ${\mathrm{log}}_{\frac{2}{3}}\frac{4}{9}$ mentally.

We ask, “To what exponent must $\frac{2}{3}$ be raised in order to get $\frac{4}{9}$ ? ” We know ${2}^{2}=4$ and ${3}^{2}=9$ , so ${\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}$ . Therefore, ${\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2$ .

tip for success

It may be tempting to use your calculator to evaluate these logarithms but try to evaluate them mentally as it will aid your understanding of what a logarithm is, and will help you navigate more complicated situations.

How To: Given a logarithm of the form $y={\mathrm{log}}_{b}\left(x\right)$ , evaluate it mentally

Rewrite the argument x as a power of b: ${b}^{y}=x$ .
Use previous knowledge of powers of b to identify y by asking, “To what exponent should b be raised in order to get x?”

Example: Solving Logarithms Mentally

Solve $y={\mathrm{log}}_{4}\left(64\right)$ without using a calculator.

Show Solution

First we rewrite the logarithm in exponential form: ${4}^{y}=64$ . Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know ${4}^{3}=64$

Therefore,

$\mathrm{log}{}_{4}\left(64\right)=3$

Try It

Solve $y={\mathrm{log}}_{121}\left(11\right)$ without using a calculator.

Show Solution

${\mathrm{log}}_{121}\left(11\right)=\frac{1}{2}$ (recall that $\sqrt{121}={\left(121\right)}^{\frac{1}{2}}=11$ )

Example: Evaluating the Logarithm of a Reciprocal

Evaluate $y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)$ without using a calculator.

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First we rewrite the logarithm in exponential form: ${3}^{y}=\frac{1}{27}$ . Next, we ask, “To what exponent must 3 be raised in order to get $\frac{1}{27}$ “?

We know ${3}^{3}=27$ , but what must we do to get the reciprocal, $\frac{1}{27}$ ? Recall from working with exponents that ${b}^{-a}=\frac{1}{{b}^{a}}$ . We use this information to write

$\begin{array}{l}{3}^{-3}=\frac{1}{{3}^{3}}=\frac{1}{27}\hfill \end{array}$

Therefore, ${\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3$ .

Try It

Evaluate $y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)$ without using a calculator.

Show Solution

{l o g}_{2} (\frac{1}{32}) = - 5

${\mathrm{log}}_{2}\left(\frac{1}{32}\right)=-5$

Using Common Logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expressions $\text{log}(x)$ means $\text{log}_{10}(x).$ We call a base- $10$ logarithm a common logarithm. Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

A General Note: Definition of the Common Logarithm

A common logarithm is a logarithm with base $10$ . We write $\text{log}_{10}(x).$ simply as $\text{log}(x)$ . The common logarithm of a positive number $x$ satisfies the following definition:

For $x>0$ ,

$y=\text{log}(x)\text{ is equivalent to }{10}^{y}=x$

We read $\text{log}(x)$ as, “the logarithm with base $10$ of $x$ ” or “the common logarithm of $x$ .”

The logarithm $y$ is the exponent to which $10$ must be raised to get $x$ .

Since the functions $y=10^{x}$ and $y=\mathrm{log}\left(x\right)$ are inverse functions, $\mathrm{log}\left({10}^{x}\right)=x$ for all $x$ and $10^{\mathrm{log}\left(x\right)}=x$ for $x>0$ .

How To: Given a common logarithm Of the form $y=\mathrm{log}\left(x\right)$ , evaluate it using a calculator+

Press [LOG].
Enter the value given for x, followed by [ ) ].
Press [ENTER].

Using Natural Logarithms

The most frequently used base for logarithms is e. Base e logarithms are important in calculus and some scientific applications; they are called natural logarithms. The base e logarithm, ${\mathrm{log}}_{e}\left(x\right)$ , has its own notation, $\mathrm{ln}\left(x\right)$ .

Most values of $\mathrm{ln}\left(x\right)$ can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, $\mathrm{ln}1=0$ . For other natural logarithms, we can use the $\mathrm{ln}$ key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e using the inverse property of logarithms.

A General Note: Definition of the Natural Logarithm

A natural logarithm is a logarithm with base e. We write ${\mathrm{log}}_{e}\left(x\right)$ simply as $\mathrm{ln}\left(x\right)$ . The natural logarithm of a positive number x satisfies the following definition:

For $x>0$ ,

$y=\mathrm{ln}\left(x\right)\text{ is equivalent to }{e}^{y}=x$

We read $\mathrm{ln}\left(x\right)$ as, “the logarithm with base e of x” or “the natural logarithm of x.”

The logarithm $y$ is the exponent to which e must be raised to get $x$ .

Since the functions $y=e^{x}$ and $y=\mathrm{ln}\left(x\right)$ are inverse functions, $\mathrm{ln}\left({e}^{x}\right)=x$ for all x and $e^{\mathrm{ln}\left(x\right)}=x$ for $x>0$ .

How To: Given a natural logarithm Of the form $y=\mathrm{ln}\left(x\right)$ , evaluate it using a calculator+

Press [LN].
Enter the value given for x, followed by [ ) ].
Press [ENTER].

Example: Evaluating a Natural Logarithm Using a Calculator

Evaluate $y=\mathrm{ln}\left(500\right)$ to four decimal places using a calculator.

Show Solution

Press [LN].
Enter 500, followed by [ ) ].
Press [ENTER].

Rounding to four decimal places, $\mathrm{ln}\left(500\right)\approx 6.2146$

Try It

Evaluate $\mathrm{ln}\left(-500\right)$ .

Show Solution

Unit 4