Find the numerical value of the log expression. {:[log a=-10quad,log b=10quad log c=11],[,log ((sqrt(ab^(5)))/(c^(9)))]:} Answer:

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Find the numerical value of the log expression.  {:[log a=-10quad,log b=10quad log c=11],[,log ((sqrt(ab^(5)))/(c^(9)))]:} Answer:

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

Find the numerical value of the log expression. $\newline$ $\begin{array}{cc} \log a=-10 \quad & \log b=10 \quad \log c=11 \\ & \log \frac{\sqrt{a b^{5}}}{c^{9}} \end{array}$ $\newline$ Answer:

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Find the numerical value of the log expression. $\newline$ $\begin{array}{cc} \log a=-10 \quad & \log b=10 \quad \log c=11 \\ & \log \frac{\sqrt{a b^{5}}}{c^{9}} \end{array}$ $\newline$ Answer: