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$Find the numerical value of the log expression. {:[log a=9quad log b=6quad log c=4],[log ((sqrt(b^(5)c^(5)))/(a^(9)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=9 \quad \log b=6 \quad \log c=4 \\ \log \frac{\sqrt{b^{5} c^{5}}}{a^{9}} \end{array}$ $\newline$ Answer:

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Math Problems

Precalculus

Quotient property of logarithms

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Q. Find the numerical value of the log expression.

\newline

\begin{array}{c} \log a=9 \quad \log b=6 \quad \log c=4 \\ \log \frac{\sqrt{b^{5} c^{5}}}{a^{9}} \end{array}

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Answer:

Identify Given Logarithm Values: Identify the given logarithm values and the expression to evaluate. $\newline$ We are given that $\log a = 9$ , $\log b = 6$ , and $\log c = 4$ . We need to find the numerical value of $\log \left(\frac{\sqrt{b^{5}c^{5}}}{a^{9}}\right)$ .
Apply Logarithm Properties: Apply the properties of logarithms to simplify the expression. $\newline$ First, we use the quotient rule of logarithms, which states that $\log(\frac{a}{b}) = \log(a) - \log(b)$ . $\newline$ $\log \left(\frac{\sqrt{b^{5}c^{5}}}{a^{9}}\right) = \log (\sqrt{b^{5}c^{5}}) - \log (a^{9})$
Simplify Square Root: Simplify the square root and apply the power rule of logarithms. $\newline$ The power rule of logarithms states that $\log(a^k) = k \cdot \log(a)$ . The square root is equivalent to raising to the power of $\frac{1}{2}$ . $\newline$ $\log (\sqrt{b^{5}c^{5}}) = \log ((b^{5}c^{5})^{\frac{1}{2}}) = \frac{1}{2} \cdot \log (b^{5}c^{5})$
Apply Product Rule: Apply the product rule of logarithms to the term inside the square root. The product rule of logarithms states that $\log(a \cdot b) = \log(a) + \log(b)$ . $(\frac{1}{2}) \cdot \log (b^{5}c^{5}) = (\frac{1}{2}) \cdot (\log (b^{5}) + \log (c^{5}))$
Apply Power Rule: Apply the power rule to the logarithms of $b$ and $c$ . $\log (b^{5}) = 5 \times \log (b)$ and $\log (c^{5}) = 5 \times \log (c)$ . $\frac{1}{2} \times (\log (b^{5}) + \log (c^{5})) = \frac{1}{2} \times (5 \times \log (b) + 5 \times \log (c))$
Substitute Given Values: Substitute the given values of $\log b$ and $\log c$ into the expression. $\newline$ $\log b = 6$ and $\log c = 4$ , so we substitute these values in. $\newline$ $(1/2) \times (5 \times \log (b) + 5 \times \log (c)) = (1/2) \times (5 \times 6 + 5 \times 4)$
Perform Arithmetic Operations: Perform the arithmetic operations. $\newline$ $(\frac{1}{2}) \times (5 \times 6 + 5 \times 4) = (\frac{1}{2}) \times (30 + 20) = (\frac{1}{2}) \times 50 = 25$
Apply Power Rule to Logarithm: Apply the power rule to the logarithm of $a^{9}$ . $\newline$ $\log (a^{9}) = 9 \times \log (a)$ and substitute the given value of $\log a = 9$ . $\newline$ $\log (a^{9}) = 9 \times 9 = 81$
Combine Results: Combine the results to find the final value of the original expression. $\log \left(\frac{\sqrt{b^{5}c^{5}}}{a^{9}}\right) = 25 - 81$
Perform Final Subtraction: Perform the final subtraction to get the numerical value. $25 - 81 = -56$