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$Find the numerical value of the log expression. {:[log a=-5quad log b=8quad log c=3],[log ((c^(3))/(a^(9)b^(8)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=-5 \quad \log b=8 \quad \log c=3 \\ \log \frac{c^{3}}{a^{9} b^{8}} \end{array}$ $\newline$ Answer:

Full solution

Q. Find the numerical value of the log expression.

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\begin{array}{c} \log a=-5 \quad \log b=8 \quad \log c=3 \\ \log \frac{c^{3}}{a^{9} b^{8}} \end{array}

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Answer:

Apply Rules of Logarithms: Apply the quotient and power rules of logarithms to the expression. $\newline$ The quotient rule of logarithms states that $\log\left(\frac{M}{N}\right) = \log(M) - \log(N)$ , and the power rule states that $\log(M^n) = n \cdot \log(M)$ . $\newline$ Using these rules, we can expand $\log \left(\frac{c^3}{a^9b^8}\right)$ as follows: $\newline$ $\log \left(\frac{c^3}{a^9b^8}\right) = \log(c^3) - \log(a^9b^8)$ $\newline$ Now apply the power rule: $\newline$ $\log(c^3) = 3 \cdot \log(c)$ $\newline$ $\log(a^9) = 9 \cdot \log(a)$ $\newline$ $\log(b^8) = 8 \cdot \log(b)$ $\newline$ So, $\log \left(\frac{c^3}{a^9b^8}\right) = 3 \cdot \log(c) - (9 \cdot \log(a) + 8 \cdot \log(b))$
Substitute Given Values: Substitute the given values of $\log(a)$ , $\log(b)$ , and $\log(c)$ into the expanded expression. $\newline$ Given: $\newline$ $\log(a) = -5$ $\newline$ $\log(b) = 8$ $\newline$ $\log(c) = 3$ $\newline$ Substitute these values into the expression: $\newline$ $3 \cdot \log(c) - (9 \cdot \log(a) + 8 \cdot \log(b)) = 3 \cdot 3 - (9 \cdot (-5) + 8 \cdot 8)$
Perform Arithmetic Operations: Perform the arithmetic operations. $\newline$ Calculate the numerical values: $\newline$ $3 \cdot 3 - (9 \cdot (-5) + 8 \cdot 8) = 9 - (-45 + 64)$ $\newline$ $9 - (-45 + 64) = 9 - (-45 + 64)$ $\newline$ $9 - (-45 + 64) = 9 + 45 - 64$ $\newline$ $9 + 45 - 64 = 54 - 64$ $\newline$ $54 - 64 = -10$