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$Find the numerical value of the log expression. {:[log a=-6quad log b=-9quad log c=-12],[log ((root(3)(b^(4)c^(7)))/(a^(4)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=-6 \quad \log b=-9 \quad \log c=-12 \\ \log \frac{\sqrt[3]{b^{4} c^{7}}}{a^{4}} \end{array}$ $\newline$ 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=-6 \quad \log b=-9 \quad \log c=-12 \\ \log \frac{\sqrt[3]{b^{4} c^{7}}}{a^{4}} \end{array}

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Answer:

Understand Given Values: Understand the given logarithmic values and the expression to evaluate. $\newline$ We are given: $\newline$ $\log a = -6$ $\newline$ $\log b = -9$ $\newline$ $\log c = -12$ $\newline$ We need to find the value of: $\newline$ $\log \left(\frac{\sqrt[3]{b^{4}c^{7}}}{a^{4}}\right)$ $\newline$ Let's start by simplifying the expression using logarithmic rules.
Apply Quotient Rule: Apply the quotient rule of logarithms to the expression. $\newline$ The quotient rule states that $\log(\frac{x}{y}) = \log(x) - \log(y)$ . $\newline$ So, $\log \left(\frac{\sqrt[3]{b^{4}c^{7}}}{a^{4}}\right) = \log \left(\sqrt[3]{b^{4}c^{7}}\right) - \log \left(a^{4}\right)$
Apply Power Rule: Apply the power rule of logarithms to the expression. $\newline$ The power rule states that $\log(x^{(n)}) = n \cdot \log(x)$ . $\newline$ So, $\log (a^{(4)}) = 4 \cdot \log(a)$ $\newline$ Since we know $\log(a) = -6$ , we can substitute to get: $\newline$ $\log (a^{(4)}) = 4 \cdot (-6) = -24$
Apply Cube Root: Apply the cube root as a power of $1/3$ to the logarithm. $\newline$ The cube root of a number can be expressed as the number raised to the power of $1/3$ . $\newline$ So, $\log (\sqrt[3]{b^{4}c^{7}}) = \log ((b^{4}c^{7})^{1/3})$ $\newline$ Using the power rule again, we get: $\newline$ $\log ((b^{4}c^{7})^{1/3}) = (1/3) \cdot \log (b^{4}c^{7})$
Apply Product Rule: Apply the product rule of logarithms to the expression. $\newline$ The product rule states that $\log(xy) = \log(x) + \log(y)$ . $\newline$ So, $(\frac{1}{3}) \cdot \log (b^{4}c^{7}) = (\frac{1}{3}) \cdot (\log (b^{4}) + \log (c^{7}))$
Apply Power Rule: Apply the power rule to the individual logarithms. $\newline$ Using the power rule again: $\newline$ $\log (b^{4}) = 4 \times \log(b)$ $\newline$ $\log (c^{7}) = 7 \times \log(c)$ $\newline$ Since we know $\log(b) = -9$ and $\log(c) = -12$ , we can substitute to get: $\newline$ $\log (b^{4}) = 4 \times (-9) = -36$ $\newline$ $\log (c^{7}) = 7 \times (-12) = -84$
Substitute Values: Substitute the values back into the expression. $\newline$ Now we have: $\newline$ (\frac{$1$}{$3$}) \times (\log (b^{$4$}) + \log (c^{$7$})) = (\frac{$1$}{$3$}) \times (\(-36 + ( $-84$ )) $\newline$ = (\frac{ $1$ }{ $3$ }) \times ( $-120$ ) $\newline$ = $-40$
Combine Results: Combine the results to find the final value of the original expression. $\newline$ We have: $\newline$ $\log \left(\frac{\sqrt[3]{b^{4}c^{7}}}{a^{4}}\right) = \log \left(\sqrt[3]{b^{4}c^{7}}\right) - \log \left(a^{4}\right)$ $\newline$ $= -40 - (-24)$ $\newline$ $= -40 + 24$ $\newline$ $= -16$