Evaluating logarithms

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: evaluate a logarithm without the calculator.

Rewrite the logarithm in exponential form: ${b}^{y}=x$ .
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

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

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

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

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

Example

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{matrix} 3^{- 3} = \frac{1}{3^{3}} = \frac{1}{27} \end{matrix}

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

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

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

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When we talked about exponential functions, we introduced the number e. Just as e was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base e is called the natural logarithmic function. The function is generally written $f(x)=ln x$ and we read it as “el en of $x . ”$

When the base of the logarithm function is 10, we call it the common logarithmic function and the base is not shown. If the base a of a logarithm is not shown, as in $f(x)=log{x}$ , we assume the base is 10.

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.

It will be important for you to use your calculator to evaluate both common and natural logarithms. Find the log and ln keys on your calculator.

Try It

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

Solution

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

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

The following video gives more examples of converting between exponential and logarithmic form as well as evaluating logarithms.

Module 9: Introduction to Exponential and Logarithmic Functions

How To: evaluate a logarithm without the calculator.

Example

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Example

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