Question 14

Write expression $\log \left(\frac{x^{17} y^{13}}{z^{4}}\right)$ as a sum or difference of logarithms with no exponents. Simplify your answer completely.
\[
\log \left(\frac{x^{17} y^{13}}{z^{4}}\right)=
\]

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Step 1 ：The given expression is a logarithm of a fraction. We can use the properties of logarithms to simplify this expression.

Step 2 ：The logarithm of a fraction can be written as the difference of the logarithms of the numerator and the denominator. So, we can rewrite \(\log \left(\frac{x^{17} y^{13}}{z^{4}}\right)\) as \(\log(x^{17} y^{13}) - \log(z^{4})\).

Step 3 ：The logarithm of a product can be written as the sum of the logarithms of the factors. So, we can rewrite \(\log(x^{17} y^{13})\) as \(\log(x^{17}) + \log(y^{13})\).

Step 4 ：The logarithm of a power can be written as the product of the exponent and the logarithm of the base. So, we can rewrite \(\log(x^{17})\) as \(17\log(x)\), \(\log(y^{13})\) as \(13\log(y)\), and \(\log(z^{4})\) as \(4\log(z)\).

Step 5 ：Substituting these expressions back into our original expression, we get \(17\log(x) + 13\log(y) - 4\log(z)\).

Step 6 ：Final Answer: \(\boxed{17 \log(x) + 13 \log(y) - 4 \log(z)}\)