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$Find the numerical value of the log expression. {:[log a=12quad log b=12quad log c=-6],[log ((a^(2))/(b^(9)c^(5)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=12 \quad \log b=12 \quad \log c=-6 \\ \log \frac{a^{2}}{b^{9} c^{5}} \end{array}$ $\newline$ Answer:

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Quotient property of logarithms

Full solution

Q. Find the numerical value of the log expression.

\newline

\begin{array}{c} \log a=12 \quad \log b=12 \quad \log c=-6 \\ \log \frac{a^{2}}{b^{9} c^{5}} \end{array}

\newline

Answer:

Apply Power Rule: Using the given values for $\log a$ , $\log b$ , and $\log c$ , we need to apply the properties of logarithms to find the value of $\log\left(\frac{a^{2}}{b^{9}c^{5}}\right)$ . First, we apply the power rule of logarithms, which states that $\log(a^n) = n \cdot \log(a)$ , to each term inside the logarithm.
Rewrite Expression: We rewrite the expression using the power rule: $\newline$ \log\left(\frac{a^{2}}{b^{9}c^{5}}\right) = \log(a^{2}) - \log(b^{9}) - \log(c^{5})\(\newline= 2 \cdot \log(a) - 9 \cdot \log(b) - 5 \cdot \log(c)\)
Substitute Given Values: Now we substitute the given values of $\log a$ , $\log b$ , and $\log c$ into the expression: $\newline$ = $2 \times 12 - 9 \times 12 - 5 \times (-6)$
Perform Arithmetic Operations: We perform the arithmetic operations: $\newline$ $= 24 - 108 + 30$
Find Value of Expression: Finally, we add and subtract the numbers to find the value of the expression: $\newline$ $= -84 + 30$ $\newline$ $= -54$

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