Full solution

Q. Find the numerical value of the log expression.

\newline

\begin{array}{c} \log a=3 \quad \log b=9 \quad \log c=-2 \\ \log \frac{\sqrt[3]{a^{5} b^{5}}}{c^{6}} \end{array}

\newline

Answer:

Apply Properties of Logarithms: Apply the properties of logarithms to the given expression. $\newline$ We have the expression $\log \left( \frac{\sqrt[3]{a^5 b^5}}{c^6} \right)$ . We can use the quotient rule of logarithms, which states that $\log \left( \frac{A}{B} \right) = \log(A) - \log(B)$ , and the power rule of logarithms, which states that $\log(A^n) = n \cdot \log(A)$ , to simplify this expression.
Apply Quotient Rule: Apply the quotient rule to the expression. $\newline$ Using the quotient rule, we get: $\newline$ $\log \left( \sqrt[3]{a^5 b^5} \right) - \log(c^6)$ .
Apply Power Rule: Apply the power rule to the expression. $\newline$ The cube root can be expressed as a power of $\frac{1}{3}$ , so we can rewrite the expression as: $\newline$ $\log \left( (a^5 b^5)^{\frac{1}{3}} \right) - \log(c^6)$ . $\newline$ Now, applying the power rule, we get: $\newline$ $\frac{1}{3} \log(a^5) + \frac{1}{3} \log(b^5) - 6 \log(c)$ .
Apply Power Rule Again: Apply the power rule again to the individual logarithms. $\newline$ We can further simplify the expression by applying the power rule to $\log(a^5)$ and $\log(b^5)$ : $\newline$ $\frac{5}{3} \log(a) + \frac{5}{3} \log(b) - 6 \log(c)$ .
Substitute Given Values: Substitute the given values of $\log(a)$ , $\log(b)$ , and $\log(c)$ into the expression. $\newline$ We are given that $\log(a) = 3$ , $\log(b) = 9$ , and $\log(c) = -2$ . Substituting these values, we get: $\newline$ $\frac{5}{3} \cdot 3 + \frac{5}{3} \cdot 9 - 6 \cdot (-2)$ .
Perform Arithmetic Operations: Perform the arithmetic operations. $\newline$ Now we calculate the numerical value: $\newline$ $\frac{5}{3} \cdot 3 = 5$ , $\newline$ $\frac{5}{3} \cdot 9 = 15$ , $\newline$ $-6 \cdot (-2) = 12$ . $\newline$ Adding these together, we get: $\newline$ $5 + 15 + 12 = 32$ .