Find the numerical value of the log expression. {:[log a=3quad log b=9quad log c=-2],[log ((root(3)(a^(5)b^(5)))/(c^(6)))]:} Answer:

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Find the numerical value of the log expression.  {:[log a=3quad log b=9quad log c=-2],[log ((root(3)(a^(5)b^(5)))/(c^(6)))]:} Answer:

Bytelearn · Accepted Answer

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

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:

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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: