Full solution

Q. Find the numerical value of the log expression.

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\begin{array}{cc} \log a=-10 \quad & \log b=10 \quad \log c=11 \\ & \log \frac{\sqrt{a b^{5}}}{c^{9}} \end{array}

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Answer:

Apply Properties of Logarithms: Apply the properties of logarithms to simplify the expression. $\newline$ We have the expression $\log \left( \frac{\sqrt{ab^5}}{c^9} \right)$ . 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 given expression. $\newline$ Using the quotient rule, we get: $\newline$ $\log \left( \sqrt{ab^5} \right) - \log \left( c^9 \right)$ .
Apply Power Rule: Apply the power rule to the terms inside the logarithms. $\newline$ The square root can be written as a power of $\frac{1}{2}$ , and we have $c^9$ , so we apply the power rule: $\newline$ $\log \left( (ab^5)^{\frac{1}{2}} \right) - 9 \cdot \log(c)$ .
Simplify First Term: Apply the power rule to the first term and simplify. $\newline$ $\frac{1}{2} \cdot \log(ab^5) - 9 \cdot \log(c)$ .
Apply Product Rule: Apply the product rule to the first term. $\newline$ The product rule states that $\log(A \cdot B) = \log(A) + \log(B)$ , so we get: $\newline$ $\frac{1}{2} \cdot (\log(a) + \log(b^5)) - 9 \cdot \log(c)$ .
Apply Power Rule to b^ $5$ : Apply the power rule to $\log(b^5)$ . $\newline$ $\frac{1}{2} \cdot (\log(a) + 5 \cdot \log(b)) - 9 \cdot \log(c)$ .
Distribute and Substitute Values: Distribute the $\frac{1}{2}$ and substitute the given values for $\log(a)$ , $\log(b)$ , and $\log(c)$ . $\newline$ $\frac{1}{2} \cdot \log(a) + \frac{5}{2} \cdot \log(b) - 9 \cdot \log(c)$ . $\newline$ Substituting the given values: $\newline$ $\frac{1}{2} \cdot (-10) + \frac{5}{2} \cdot 10 - 9 \cdot 11$ .
Perform Arithmetic Operations: Perform the arithmetic operations. $\newline$ $-5 + 25 - 99$ .
Combine Terms: Combine the terms to find the final numerical value. $\newline$ $-5 + 25 - 99 = 20 - 99 = -79$ .