Evaluate Logarithms

Learning Objectives

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, ${\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2$ .

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 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.

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.

Solution

{l o g}_{121} (11) = \frac{1}{2}

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

\sqrt{121} = {(121)}^{\frac{1}{2}} = 11

$\sqrt{121}={\left(121\right)}^{\frac{1}{2}}=11$ )

Example: Evaluating the Logarithm of a Reciprocal

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

Solution

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

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

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.

Solution

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

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

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Use common logarithms

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b, exponent x, and output y. Then we write $x={\mathrm{log}}_{b}\left(y\right)$ .

Example: Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

${2}^{3}=8$
${5}^{2}=25$
${10}^{-4}=\frac{1}{10,000}$

Solution

First, identify the values of b, y, and x. Then, write the equation in the form

x = {l o g}_{b} (y)

$x={\mathrm{log}}_{b}\left(y\right)$ .

${2}^{3}=8$ Here, b = 2, x = 3, and y = 8. Therefore, the equation ${2}^{3}=8$ is equivalent to ${\mathrm{log}}_{2}\left(8\right)=3$ .
${5}^{2}=25$ Here, b = 5, x = 2, and y = 25. Therefore, the equation ${5}^{2}=25$ is equivalent to ${\mathrm{log}}_{5}\left(25\right)=2$ .
${10}^{-4}=\frac{1}{10,000}$ Here, b = 10, x = –4, and $y=\frac{1}{10,000}$ . Therefore, the equation ${10}^{-4}=\frac{1}{10,000}$ is equivalent to ${\text{log}}_{10}\left(\frac{1}{10,000}\right)=-4$ .

Try It

Write the following exponential equations in logarithmic form.

${3}^{2}=9$
${5}^{3}=125$
${2}^{-1}=\frac{1}{2}$

Solution

${3}^{2}=9$ is equivalent to ${\mathrm{log}}_{3}\left(9\right)=2$
${5}^{3}=125$ is equivalent to ${\mathrm{log}}_{5}\left(125\right)=3$
${2}^{-1}=\frac{1}{2}$ is equivalent to ${\text{log}}_{2}\left(\frac{1}{2}\right)=-1$

Use 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 with 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.

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

Solution

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Exponential and Logarithmic Functions

Learning Objectives

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

Example: Solving Logarithms Mentally

Try It

Example: Evaluating the Logarithm of a Reciprocal

Try It

Use common logarithms

Example: Converting from Exponential Form to Logarithmic Form

Try It

Use natural logarithms

A General Note: Definition of the Natural Logarithm

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

Example: Evaluating a Natural Logarithm Using a Calculator

Try It

Candela Citations

Exponential and Logarithmic Functions