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Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log x,log y, and
log z.

log ((sqrt(x^(5)))/(y^(3)z^(4)))
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{\sqrt{x^{5}}}{y^{3} z^{4}}$ $\newline$ Answer:

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Evaluate definite integrals using the power rule

Full solution

Q. Expand the logarithm fully using the properties of logs. Express the final answer in terms of

\log x, \log y

, and

\log z

.

\newline

\log \frac{\sqrt{x^{5}}}{y^{3} z^{4}}

\newline

Answer:

Apply Quotient Rule: Apply the quotient rule for logarithms, which states that $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ . $\log\left(\frac{\sqrt{x^{5}}}{y^{3}z^{4}}\right) = \log(\sqrt{x^{5}}) - \log(y^{3}z^{4})$
Apply Product Rule: Apply the product rule for logarithms, which states that $\log(ab) = \log(a) + \log(b)$ , to the denominator of the original expression. $\newline$ $\log(y^{3}z^{4}) = \log(y^{3}) + \log(z^{4})$
Apply Power Rule: Apply the power rule for logarithms, which states that $\log(a^n) = n\log(a)$ , to each logarithm. $\newline$ $\log(\sqrt{x^{5}}) = \frac{1}{2} \cdot \log(x^{5}) = \frac{5}{2} \cdot \log(x)$ $\newline$ $\log(y^{3}) = 3 \cdot \log(y)$ $\newline$ $\log(z^{4}) = 4 \cdot \log(z)$
Substitute Results: Substitute the results from Step $3$ back into the expression from Step $1$ . \log\left(\frac{\sqrt{x^{\(5\)}}}{y^{\(3\)}z^{\(4\)}}\right) = \left(\frac{\(5\)}{\(2\)}\right) \cdot \log(x) - \left(\(3 \cdot \log(y) + $4$ \cdot \log(z)\right)
Distribute Negative Sign: Distribute the negative sign to both terms in the parentheses. $\newline$ $\log\left(\frac{\sqrt{x^{5}}}{y^{3}z^{4}}\right) = \frac{5}{2} \cdot \log(x) - 3 \cdot \log(y) - 4 \cdot \log(z)$

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