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$Find the numerical value of the log expression. {:[log a=8quad log b=2quad log c=-4],[log ((c^(8))/(a^(9)b^(9)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=8 \quad \log b=2 \quad \log c=-4 \\ \log \frac{c^{8}}{a^{9} b^{9}} \end{array}$ $\newline$ 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=8 \quad \log b=2 \quad \log c=-4 \\ \log \frac{c^{8}}{a^{9} b^{9}} \end{array}

\newline

Answer:

Apply Quotient Rule: We are given the values of $\log a$ , $\log b$ , and $\log c$ , and we need to find the value of $\log\left(\frac{c^{8}}{a^{9}b^{9}}\right)$ . We can use the properties of logarithms to simplify the expression. $\newline$ First, let's apply the quotient rule of logarithms, which states that $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ . $\newline$ $\log\left(\frac{c^{8}}{a^{9}b^{9}}\right) = \log(c^{8}) - \log(a^{9}b^{9})$
Apply Product Rule: Now, let's apply the product rule of logarithms to the second term, which states that $\log(ab) = \log(a) + \log(b)$ . $\newline$ $\log(a^{9}b^{9}) = \log(a^{9}) + \log(b^{9})$
Apply Power Rule: Next, we apply the power rule of logarithms, which states that $\log(a^n) = n\log(a)$ , to each term. $\newline$ $\log(c^{8}) = 8\log(c)$ $\newline$ $\log(a^{9}) = 9\log(a)$ $\newline$ $\log(b^{9}) = 9\log(b)$
Substitute Given Values: Now we substitute the given values of $\log a$ , $\log b$ , and $\log c$ into the expression. $\newline$ $\log(c^{8}) - (\log(a^{9}) + \log(b^{9})) = 8\cdot\log(c) - (9\cdot\log(a) + 9\cdot\log(b))$ $\newline$ $= 8\cdot(-4) - (9\cdot8 + 9\cdot2)$
Perform Arithmetic Operations: We perform the arithmetic operations. $\newline$ $= -32 - (72 + 18)$ $\newline$ $= -32 - 90$ $\newline$ $= -122$

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