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Find the numerical value of the log expression. {:[log a=10quad log b=12quad log c=-3],[log ((a^(2)b^(9))/(root(3)(c^(5))))]:} Answer:

Find the numerical value of the log expression.\newlinelog⁡a=10log⁡b=12log⁡c=−3log⁡a2b9c53 \begin{array}{c} \log a=10 \quad \log b=12 \quad \log c=-3 \\ \log \frac{a^{2} b^{9}}{\sqrt[3]{c^{5}}} \end{array} \newlineAnswer:

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Q. Find the numerical value of the log expression.\newlinelog⁡a=10log⁡b=12log⁡c=−3log⁡a2b9c53 \begin{array}{c} \log a=10 \quad \log b=12 \quad \log c=-3 \\ \log \frac{a^{2} b^{9}}{\sqrt[3]{c^{5}}} \end{array} \newlineAnswer:
  1. Use Logarithm Properties: Use the properties of logarithms to simplify the expression. The properties we will use are the power rule log⁡(xy)=ylog⁡(x)\log(x^y) = y\log(x), the quotient rule log⁡(xy)=log⁡(x)−log⁡(y)\log\left(\frac{x}{y}\right) = \log(x) - \log(y), and the fact that log⁡(xn)=log⁡(x1/n)\log(\sqrt[n]{x}) = \log(x^{1/n}).
    \log\left(\frac{a^{2}b^{9}}{c^{\frac{1}{3}}^{5}}\right) = 2\log(a) + 9\log(b) - 5\log(c^{\frac{1}{3}})
  2. Apply Power Rule: Apply the power rule to log⁡(c13)\log(c^{\frac{1}{3}}).\newlinelog⁡(c13)=13log⁡(c)\log(c^{\frac{1}{3}}) = \frac{1}{3}\log(c)
  3. Substitute Given Values: Substitute the given values for log⁡(a)\log(a), log⁡(b)\log(b), and log⁡(c)\log(c) into the expression.2⋅log⁡(a)+9⋅log⁡(b)−5⋅log⁡(c13)=2⋅10+9⋅12−5⋅(13⋅(−3))2\cdot\log(a) + 9\cdot\log(b) - 5\cdot\log(c^{\frac{1}{3}}) = 2\cdot 10 + 9\cdot 12 - 5\cdot\left(\frac{1}{3}\cdot(-3)\right)
  4. Perform Arithmetic Operations: Perform the arithmetic operations. 2×10+9×12−5×((1/3)×(−3))=20+108−5×(−1)=20+108+5=128+5=1332\times10 + 9\times12 - 5\times((1/3)\times(-3)) = 20 + 108 - 5\times(-1) = 20 + 108 + 5 = 128 + 5 = 133

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