Question

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where​ possible, evaluate logarithmic expressions.
one half left parenthesis log Subscript 4 Baseline x plus log Subscript 4 Baseline y right parenthesis minus 5 log Subscript 4 Baseline left parenthesis x plus 5 right parenthesis
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Part 1
one half left parenthesis log Subscript 4 Baseline x plus log Subscript 4 Baseline y right parenthesis minus 5 log Subscript 4 Baseline left parenthesis x plus 5 right parenthesis
​=
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To condense the given expression \( \frac{1}{2} (\log_4 x + \log_4 y) - 5 \log_4 (x + 5) \), we can use properties of logarithms.

1. **Use the Addition Property of Logarithms**:
\[
\log_a b + \log_a c = \log_a (bc)
\]
Thus, we can condense \( \log_4 x + \log_4 y \) as follows:
\[
\frac{1}{2} (\log_4 x + \log_4 y) = \frac{1}{2} \log_4 (xy)
\]

2. **Use the Power Property of Logarithms**:
\[
n \log_a b = \log_a (b^n)
\]
So, we can write:
\[
-5 \log_4 (x + 5) = \log_4 ((x + 5)^{-5})
\]

3. **Combine the Logarithms**:
Now we can combine the two logarithmic expressions:
\[
\frac{1}{2} \log_4 (xy) - 5 \log_4 (x + 5) = \frac{1}{2} \log_4 (xy) + \log_4 ((x + 5)^{-5})
\]
We can apply the addition property again:
\[
= \log_4 \left( (xy)^{1/2} \cdot (x + 5)^{-5} \right)
\]

4. **Final Expression**:
Combine everything to write the final expression as a single logarithm:
\[
= \log_4 \left( \frac{\sqrt{xy}}{(x + 5)^5} \right)
\]

Thus, the condensed expression is:
\[
\log_4 \left( \frac{\sqrt{xy}}{(x + 5)^5} \right)
\]