Which of the following rewritten logarithms uses the Power Rule of Logarithms to evaluate log10^4/5

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1 answer

To evaluate \( \log\left(\frac{10^4}{5}\right) \) using the Power Rule of Logarithms, we first apply the Quotient Rule:

\[ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \]

So, we can rewrite the logarithm as:

\[ \log\left(\frac{10^4}{5}\right) = \log(10^4) - \log(5) \]

Next, we apply the Power Rule of Logarithms, which states:

\[ \log(a^b) = b \cdot \log(a) \]

Using this rule on \( \log(10^4) \):

\[ \log(10^4) = 4 \cdot \log(10) \]

Since \( \log(10) = 1 \) in base 10, we have:

\[ \log(10^4) = 4 \cdot 1 = 4 \]

Putting it all together, we get:

\[ \log\left(\frac{10^4}{5}\right) = 4 - \log(5) \]

Thus, the rewritten logarithm that uses the Power Rule of Logarithms is:

\[ \log\left(\frac{10^4}{5}\right) = 4 - \log(5) \]