Question

Use properties of logarithms to expand the logarithmic expression as much as possible. Assume s and t are positive real numbers.
log RootIndex 5 StartRoot StartFraction s Over t EndFraction EndRoot
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Part 1
log RootIndex 5 StartRoot StartFraction s Over t EndFraction EndRoot
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​(Type an exact answer in simplified​ form.)

Answer 1

To expand the logarithmic expression \(\log_5\left(\sqrt{\frac{s}{t}}\right)\), we can use the properties of logarithms, specifically the power rule and the quotient rule.

1. **Power Rule**: \(\log_b(a^c) = c \cdot \log_b(a)\)
2. **Quotient Rule**: \(\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\)

Starting with the given expression:

\[
\log_5\left(\sqrt{\frac{s}{t}}\right)
\]

We can rewrite the square root as a power:

\[
\sqrt{\frac{s}{t}} = \left(\frac{s}{t}\right)^{1/2}
\]

Now, applying the power rule:

\[
\log_5\left(\sqrt{\frac{s}{t}}\right) = \log_5\left(\left(\frac{s}{t}\right)^{1/2}\right) = \frac{1}{2} \log_5\left(\frac{s}{t}\right)
\]

Next, we apply the quotient rule:

\[
\log_5\left(\frac{s}{t}\right) = \log_5(s) - \log_5(t)
\]

Substituting this back into our expression gives:

\[
\frac{1}{2} \log_5\left(\frac{s}{t}\right) = \frac{1}{2} \left(\log_5(s) - \log_5(t)\right)
\]

Thus, the full expanded form is:

\[
\frac{1}{2} \log_5(s) - \frac{1}{2} \log_5(t)
\]

Putting it all together, we have:

\[
\log_5\left(\sqrt{\frac{s}{t}}\right) = \frac{1}{2} \log_5(s) - \frac{1}{2} \log_5(t)
\]

So, the final answer is:

\[
\frac{1}{2} \log_5(s) - \frac{1}{2} \log_5(t)
\]