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Exercises Exponentials and logarithms

Example 4 from Algebra and Trigonometry, 6.3 Logarithmic Functions

Evaluating the Logarithm of a Reciprocal

Evaluate $y = \log_{3} (\frac{1}{27})$ without using a calculator.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

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

$\begin{array}{l} 3^{- 3} = \frac{1}{3^{3}} \\ = \frac{1}{27} \end{array}$
Therefore, $\log_{3} (\frac{1}{27}) = - 3.$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Evaluate $y = \log_{2} (\frac{1}{32})$ without using a calculator.

How did it go?

PreviousSolving Logarithms Mentally Next Finding the Value of a Common Logarithm Mentally

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 6.3, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.