To apply the Power Rule of Logarithms, we usually look at logarithmic expressions of the form \(\log_b(a^c) = c \cdot \log_b(a)\). However, in this case, we are evaluating \(\log_{10}(4/5)\) and we need to look for the appropriate simplifications.
The given options suggest manipulations of that logarithm, but none of them directly apply the Power Rule, since \( \frac{4}{5} \) cannot be treated as a power of something.
Instead, the logarithm of a fraction can be expressed using the Quotient Rule of Logarithms, which states that: \[ \log_b \left(\frac{a}{c}\right) = \log_b(a) - \log_b(c) \]
Applying this to \(\log_{10}(4/5)\), we get: \[ \log_{10}(4/5) = \log_{10}(4) - \log_{10}(5) \]
Thus, this expression does not involve the Power Rule directly, but based on the options you provided, it seems that I need to determine which one involves some form of manipulation relating to the logarithm of \( \frac{4}{5} \).
Let's rewrite \(\log_{10}(4/5)\) as follows to see if it matches one of your options:
- \(\log_{10}(4/5) = -\log_{10}(5/4)\) (because \(\frac{4}{5}\) can be expressed as the negative of the logarithm of its reciprocal).
Looking at your options:
- The expression \(-\frac{5}{4} \log_{10}\left(\frac{4}{5}\right)\) also won't be the answer since it doesn't correctly represent the logarithm as shown, and none of the responses using powers directly relate.
However, based on the give options, if we interpret it as:
- \(-5 \log_{10}\left(\frac{4}{5}\right)\) leads directly to a negative and doesn't utilize the Power Rule, and
- \(-\log_{10}\left(\frac{5}{4}\right)\) or comparison isn't entirely present here.
Thus evaluating your choices purely for expressions doesn't yield results befitting those criteria fully.
The most relatable choice to \(\log_{10}(4/5)\) as simplified could lead us to -\(\log_{10}(5/4)\).
If I had to choose based on your options without hearing further explanations on the constructs: The closest and most directly imply connected expression remains: \[
- \log_{10}\left(\frac{5}{4}\right) \]
So, my final suggestion from those would choose to relate to \(-4\log_{10}\left(\frac{4}{5}\right)\) along my explanations.
If you have any reconfigurations or deeper context, that could also help clarify to help provide a precise response accordingly.
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