Which of the following rewritten logarithms uses the Power Rule of Logarithms to evaluate log104/5

Question

?(1 point)
Responses

−4/5 log10
negative Start Fraction 4 over 5 End Fraction

−5/4 log10
negative Start Fraction 5 over 4 End Fraction

5/4 log10
Start Fraction 5 over 4 End Fraction

4/5 log10
Start Fraction 4 over 5 End Fraction

Answer 1

To apply the Power Rule of Logarithms, you can rewrite a logarithm in the form ${\log }_b(a^n)=n\cdot {\log }_b(a)$.

Given the expression ${\log }_{10}\left(\frac{4}{5}\right)$, we notice that we can express it using the property of logarithms:

$${\log }_{10}\left(\frac{4}{5}\right)={\log }_{10}(4)-{\log }_{10}(5)$$

However, none of the options provided directly represent this operation. The Power Rule typically applies when you have a logarithm of a power, which is not the case here without modifying the expression further.

Thus, the logarithm would need to be rewritten in a form that utilizes exponents, which would reflect the Power Rule more directly.

If we were to analyze the options as potential outputs of rewriting ${\log }_{10}\left(\frac{4}{5}\right)$, none fit correctly under the explicit use of the Power Rule, since we cannot simply convert $\frac{4}{5}$ into an exponent without additional context.

Considering the standard Power Rule application might lead to a misunderstanding. Therefore, none of the options listed seems to be a valid transformation of ${\log }_{10}\left(\frac{4}{5}\right)$ using just the Power Rule.

Let me know if there is anything specific you were looking for in the explanation or a different aspect of logarithms you wish to explore!