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$Find the numerical value of the log expression. {:[log a=11quad log b=-7quad log c=-12],[log ((root(3)(c))/(a^(9)b^(6)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=11 \quad \log b=-7 \quad \log c=-12 \\ \log \frac{\sqrt[3]{c}}{a^{9} b^{6}} \end{array}$ $\newline$ 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=11 \quad \log b=-7 \quad \log c=-12 \\ \log \frac{\sqrt[3]{c}}{a^{9} b^{6}} \end{array}

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Answer:

Given Log Values: We are given the values of $\log a = 11$ , $\log b = -7$ , and $\log c = -12$ . We need to find the value of $\log \left( \frac{\sqrt[3]{c}}{a^{9}b^{6}} \right)$ . $\newline$ First, let's apply the logarithm rules to simplify the expression. $\newline$ Using the quotient rule of logarithms, which states that $\log \left( \frac{x}{y} \right) = \log x - \log y$ , we can write: $\newline$ $\log \left( \frac{\sqrt[3]{c}}{a^{9}b^{6}} \right) = \log \left( \sqrt[3]{c} \right) - \log \left( a^{9}b^{6} \right)$ .
Simplify Expression: Next, we apply the power rule of logarithms, which states that $\log \left( x^{n} \right) = n \cdot \log x$ , to the terms $a^{9}$ and $b^{6}$ : $\newline$ $\log \left( a^{9} \right) = 9 \cdot \log a$ and $\log \left( b^{6} \right) = 6 \cdot \log b$ . $\newline$ We also apply the same rule to $\sqrt[3]{c}$ , which can be written as $c^{1/3}$ , so $\log \left( \sqrt[3]{c} \right) = \frac{1}{3} \cdot \log c$ .
Apply Log Rules: Now, we substitute the given logarithm values into the expression: $\newline$ $\log \left( \sqrt[3]{c} \right) - \log \left( a^{9}b^{6} \right)$ becomes $\frac{1}{3} \cdot \log c - (9 \cdot \log a + 6 \cdot \log b)$ . $\newline$ Substituting the values, we get $\frac{1}{3} \cdot (-12) - (9 \cdot 11 + 6 \cdot (-7))$ .
Substitute Values: Let's perform the calculations: $\newline$ $\frac{1}{3} \cdot (-12) = -4$ , $\newline$ $9 \cdot 11 = 99$ , $\newline$ and $6 \cdot (-7) = -42$ . $\newline$ So, the expression becomes $-4 - (99 - 42)$ .
Perform Calculations: Finally, we calculate the remaining operation: $\newline$ $-4 - (99 - 42) = -4 - 99 + 42 = -4 - 57 = -61$ . $\newline$ Therefore, the numerical value of the log expression is $-61$ .