Question
Write the expression as a single logarithm. \[ 3 \log _{8}(z-3)-5 \log _{8}(z+6) \] log ( \( \square \),
Ask by Gough Mitchell. in the United States
Mar 16,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( \log_{8}\left(\frac{(z-3)^3}{(z+6)^5}\right) \).
Solution
We begin with the expression
\[
3 \log_{8}(z-3) - 5 \log_{8}(z+6).
\]
Using the logarithm rule \( a\log_b c = \log_b(c^a) \), we can rewrite each term:
\[
3 \log_{8}(z-3) = \log_{8}((z-3)^3),
\]
\[
5 \log_{8}(z+6) = \log_{8}((z+6)^5).
\]
Then, using the difference property of logarithms
\[
\log_b A - \log_b B = \log_b\left(\frac{A}{B}\right),
\]
we combine the two logarithms:
\[
\log_{8}((z-3)^3) - \log_{8}((z+6)^5) = \log_{8}\left(\frac{(z-3)^3}{(z+6)^5}\right).
\]
Thus, the original expression as a single logarithm is:
\[
\log_{8}\left(\frac{(z-3)^3}{(z+6)^5}\right).
\]
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Simplify this solution Extra Insights
To combine the expression \( 3 \log_{8}(z-3) - 5 \log_{8}(z+6) \) into a single logarithm, we first apply the power rule and then the quotient rule of logarithms. We have: \[ 3 \log_{8}(z-3) = \log_{8}((z-3)^3) \] \[ -5 \log_{8}(z+6) = \log_{8}((z+6)^{-5}) \] Now, we can combine these logs: \[ \log_{8}((z-3)^3) + \log_{8}((z+6)^{-5}) = \log_{8}\left(\frac{(z-3)^3}{(z+6)^5}\right) \] Thus, the expression simplifies to: \[ \log_{8}\left(\frac{(z-3)^3}{(z+6)^5}\right) \]
