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Exercises Exponentials and logarithms

Example 11 from Algebra and Trigonometry, 6.5 Logarithmic Properties

Rewriting as a Single Logarithm

Rewrite $2 \log x - 4 \log (x + 5) + \frac{1}{x} \log (3 x + 5)$ as a single logarithm.

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
We apply the power rule first:

$2 \log x - 4 \log (x + 5) + \frac{1}{x} \log (3 x + 5) = \log (x^{2}) - \log {(x + 5)}^{4} + \log ({(3 x + 5)}^{x^{- 1}})$
Next we rearrange and apply the product rule to the sum:

$\log (x^{2}) - \log {(x + 5)}^{4} + \log ({(3 x + 5)}^{x^{- 1}})$
$= \log (x^{2}) + \log ({(3 x + 5)}^{x^{- 1}}) - \log {(x + 5)}^{4}$
$= \log (x^{2} {(3 x + 5)}^{x^{- 1}}) - \log {(x + 5)}^{4}$
Finally, we apply the quotient rule to the difference:

$= \log (x^{2} {(3 x + 5)}^{x^{−1}}) - {\log (x + 5)}^{4} = \log \frac{x^{2} {(3 x + 5)}^{x^{−1}}}{{(x + 5)}^{4}}$

Your turn

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

Condense $4 (3 \log (x) + \log (x + 5) - \log (2 x + 3)) .$

How did it go?

PreviousCondensing Complex Logarithmic Expressions Next Applying of the Laws of Logs

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 6.5, 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.