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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)))/(z^(3)y^(2)))
Answer:

Expand the logarithm fully using the properties of logs. Express the final answer in terms of $\log x, \log y$ , and $\log z$ . $\newline$ $\log \frac{\sqrt{x^{5}}}{z^{3} y^{2}}$ $\newline$ 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}}}{z^{3} y^{2}}

\newline

Answer:

Apply Quotient Rule: We start by applying the quotient rule of logarithms, which states that $\log(\frac{a}{b}) = \log(a) - \log(b)$ . In this case, we have $a = \sqrt{x^5}$ and $b = z^3y^2$ .
Apply Power Rule: Next, we apply the power rule of logarithms, which states that $\log(a^b) = b\cdot\log(a)$ . For the numerator, we have $\sqrt{x^5}$ which is $x^{5/2}$ , and for the denominator, we have $z^3$ and $y^2$ .
Rewrite Using Product Rule: We can now rewrite the logarithm as $\log(x^{\frac{5}{2}}) - (\log(z^3) + \log(y^2))$ by applying the product rule of logarithms, which states that $\log(ab) = \log(a) + \log(b)$ , to the denominator.
Apply Power Rule: We apply the power rule to each term: $\log(x^{\frac{5}{2}})$ becomes $\frac{5}{2}\cdot\log(x)$ , $\log(z^3)$ becomes $3\cdot\log(z)$ , and $\log(y^2)$ becomes $2\cdot\log(y)$ .
Subtract Logs: Subtracting the logs in the denominator from the log in the numerator, we get $(\frac{5}{2})\log(x) - 3\log(z) - 2\log(y)$ .
Final Expanded Form: This is the fully expanded form of the original logarithm in terms of $\log x$ , $\log y$ , and $\log z$ .

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