Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.
log Subscript 5 Baseline left parenthesis StartFraction 5 Over y EndFraction right parenthesis
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Part 1
log Subscript 5 Baseline left parenthesis StartFraction 5 Over y EndFraction right parenthesis
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Answer 1

To expand the logarithmic expression \(\log_5\left(\frac{5}{y}\right)\) using properties of logarithms, we can use the quotient rule, which states that \(\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\).

Applying the quotient rule, we get:

\[
\log_5\left(\frac{5}{y}\right) = \log_5(5) - \log_5(y)
\]

Next, we can simplify \(\log_5(5)\). Since \(5\) is the base of the logarithm, \(\log_5(5) = 1\).

Thus, we have:

\[
\log_5\left(\frac{5}{y}\right) = 1 - \log_5(y)
\]

So the final expression after expanding is:

\[
\boxed{1 - \log_5(y)}
\]