Full solution

Q. Find the numerical value of the log expression.

\newline

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

\newline

Answer:

Apply Logarithm Power Rule: Apply the logarithm power rule to the expression. $\newline$ The power rule of logarithms states that $\log(a^n) = n \times \log(a)$ . Let's apply this rule to each term in the expression. $\newline$ $\log(a^{9}) = 9 \times \log(a)$ $\newline$ $\log(b^{9}) = 9 \times \log(b)$ $\newline$ $\log(c^{8}) = 8 \times \log(c)$
Substitute Given Log Values: Substitute the given logarithm values into the expression. $\newline$ We have been given $\log a = -8$ , $\log b = 9$ , and $\log c = -3$ . Let's substitute these values into the expression from Step $1$ . $\newline$ $\log(a^{9}) = 9 \times (-8)$ $\newline$ $\log(b^{9}) = 9 \times 9$ $\newline$ $\log(c^{8}) = 8 \times (-3)$
Calculate Values: Calculate the values from Step $2$ . $\newline$ Now we calculate the numerical values for each term. $\newline$ $\log(a^{9}) = 9 \times (-8) = -72$ $\newline$ $\log(b^{9}) = 9 \times 9 = 81$ $\newline$ $\log(c^{8}) = 8 \times (-3) = -24$
Apply Logarithm Quotient Rule: Apply the logarithm quotient rule to the original expression. $\newline$ The quotient rule of logarithms states that $\log(\frac{a}{b}) = \log(a) - \log(b)$ . Let's apply this rule to the original expression. $\newline$ $\log(\frac{a^{9}b^{9}}{c^{8}}) = \log(a^{9}) + \log(b^{9}) - \log(c^{8})$
Substitute Calculated Values: Substitute the calculated values from Step $3$ into the expression from Step $4$ . $\newline$ Now we substitute the values we found into the expression. $\newline$ $\log\left(\frac{a^{9}b^{9}}{c^{8}}\right) = (-72) + 81 - (-24)$
Calculate Final Value: Calculate the final numerical value. $\newline$ Now we perform the arithmetic to find the numerical value. $\newline$ $\log\left(\frac{a^{9}b^{9}}{c^{8}}\right) = (-72) + 81 + 24$ $\newline$ $\log\left(\frac{a^{9}b^{9}}{c^{8}}\right) = 9 + 24$ $\newline$ $\log\left(\frac{a^{9}b^{9}}{c^{8}}\right) = 33$