Question
Condense the expression to a single logarithm. Write fractional exponents as radicals. Assume that all variables represent positive numbers. \( 6 \log _{2}(x+7)-\log _{2}(x-1)-\log _{2}(x-5)=\square \) \( > \) Next Question
Ask by Christensen Watkins. in the United States
Mar 15,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The condensed expression is \( \log_{2}\left(\frac{(x+7)^6}{(x-1)(x-5)}\right) \).
Solution
\( 6 \log_{2}(x+7) = \log_{2}((x+7)^6) \)
Thus, the expression becomes:
\[
\log_{2}((x+7)^6) - \log_{2}(x-1) - \log_{2}(x-5)
\]
Using the logarithm quotient rule:
\[
\log_{2}((x+7)^6) - \log_{2}((x-1)(x-5)) = \log_{2}\left(\frac{(x+7)^6}{(x-1)(x-5)}\right)
\]
So, the condensed expression is:
\[
\log_{2}\left(\frac{(x+7)^6}{(x-1)(x-5)}\right)
\]
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Simplify this solution Beyond the Answer
To condense the expression \( 6 \log _{2}(x+7)-\log _{2}(x-1)-\log _{2}(x-5) \), we can start by applying the property \( a \log_b(m) = \log_b(m^a) \). This gives us: \[ \log_{2}((x+7)^6) - \log_{2}(x-1) - \log_{2}(x-5) \] Now we can use the property \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \) to combine the logs: \[ \log_{2}\left(\frac{(x+7)^6}{(x-1)(x-5)}\right) \] So, the entire expression condenses to: \[ \log_{2}\left(\frac{(x+7)^6}{(x-1)(x-5)}\right) \] And that's the final single logarithm!
