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$Find the numerical value of the log expression. {:[log a=4quad log b=-8quad log c=-3],[log ((sqrt(a^(3)b^(5)))/(c^(5)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=4 \quad \log b=-8 \quad \log c=-3 \\ \log \frac{\sqrt{a^{3} b^{5}}}{c^{5}} \end{array}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Find the numerical value of the log expression.

\newline

\begin{array}{c} \log a=4 \quad \log b=-8 \quad \log c=-3 \\ \log \frac{\sqrt{a^{3} b^{5}}}{c^{5}} \end{array}

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Answer:

Identify Given Values: Identify the given logarithmic values and the expression to be evaluated. $\newline$ We are given: $\newline$ $\log a = 4$ $\newline$ $\log b = -8$ $\newline$ $\log c = -3$ $\newline$ We need to find the value of $\log \left(\frac{\sqrt{a^{3}b^{5}}}{c^{5}}\right)$ .
Apply Logarithmic Properties: Apply the properties of logarithms to the expression. $\newline$ Using the quotient rule of logarithms, which states that $\log(\frac{x}{y}) = \log(x) - \log(y)$ , we can write: $\newline$ $\log \left(\frac{\sqrt{a^{3}b^{5}}}{c^{5}}\right) = \log (\sqrt{a^{3}b^{5}}) - \log (c^{5})$
Apply Power Rule: Apply the power rule of logarithms to the expression inside the square root. $\newline$ The power rule states that $\log(x^k) = k \cdot \log(x)$ . Since we have a square root, which is equivalent to the power of $\frac{1}{2}$ , we can write: $\newline$ $\log (\sqrt{a^{3}b^{5}}) = \left(\frac{1}{2}\right) \cdot \log (a^{3}b^{5})$
Apply Product Rule: Apply the product rule of logarithms to the expression inside the logarithm. $\newline$ The product rule states that $\log(xy) = \log(x) + \log(y)$ . Therefore: $\newline$ $(\frac{1}{2}) \cdot \log (a^{3}b^{5}) = (\frac{1}{2}) \cdot (\log (a^{3}) + \log (b^{5}))$
Apply Power Rule: Apply the power rule to the individual logarithms. $\newline$ $\log (a^{3}) = 3 \cdot \log (a)$ $\newline$ $\log (b^{5}) = 5 \cdot \log (b)$ $\newline$ So, $(1/2) \cdot (\log (a^{3}) + \log (b^{5})) = (1/2) \cdot (3 \cdot \log (a) + 5 \cdot \log (b))$
Substitute Given Values: Substitute the given values of $\log a$ and $\log b$ into the expression. $\log a = 4$ and $\log b = -8$ , so: $\frac{1}{2} \times (3 \times \log (a) + 5 \times \log (b)) = \frac{1}{2} \times (3 \times 4 + 5 \times (-8))$
Perform Arithmetic Operations: Perform the arithmetic operations. $\newline$ (\frac{$1$}{$2$}) \times (\(3 \times $4$ + $5$ \times ( $-8$ )) = (\frac{ $1$ }{ $2$ }) \times ( $12$ - $40$ ) = (\frac{ $1$ }{ $2$ }) \times ( $-28$ ) = $-14$
Apply Power Rule: Apply the power rule to the logarithm of $c^{5}$ . $\newline$ $\log (c^{5}) = 5 \cdot \log (c)$ $\newline$ Substitute the given value of $\log c$ : $\newline$ $5 \cdot \log (c) = 5 \cdot (-3)$
Perform Arithmetic Operation: Perform the arithmetic operation for $\log (c^{5})$ . $5 \times (-3) = -15$
Combine Results: Combine the results from Step $7$ and Step $9$ using the result from Step $2$ . $\newline$ $\log \left(\frac{\sqrt{a^{3}b^{5}}}{c^{5}}\right) = -14 - (-15)$
Perform Final Arithmetic: Perform the final arithmetic operation to find the numerical value. $\newline$ $-14 - (-15) = -14 + 15 = 1$