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

Find the numerical value of the log expression.\newlinelog⁡a=9log⁡b=12log⁡c=3log⁡a3c3b23 \begin{array}{c} \log a=9 \quad \log b=12 \quad \log c=3 \\ \log \frac{a^{3} c^{3}}{\sqrt[3]{b^{2}}} \end{array} \newlineAnswer:

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Q. Find the numerical value of the log expression.\newlinelog⁡a=9log⁡b=12log⁡c=3log⁡a3c3b23 \begin{array}{c} \log a=9 \quad \log b=12 \quad \log c=3 \\ \log \frac{a^{3} c^{3}}{\sqrt[3]{b^{2}}} \end{array} \newlineAnswer:
  1. Rewrite with given logs: Use the given logarithmic values to rewrite the expression.\newlineWe are given log⁡(a)=9\log(a) = 9, log⁡(b)=12\log(b) = 12, and log⁡(c)=3\log(c) = 3. We need to evaluate log⁡(a3c3b23)\log\left(\frac{a^{3}c^{3}}{\sqrt[3]{b^{2}}}\right).
  2. Simplify using properties: Apply the logarithmic properties to simplify the expression.\newlineThe properties we will use are:\newline11. log⁡(xk)=k⋅log⁡(x)\log(x^k) = k \cdot \log(x) for any real number kk.\newline22. log⁡(xy)=log⁡(x)−log⁡(y)\log\left(\frac{x}{y}\right) = \log(x) - \log(y) for any positive real numbers xx and yy.\newline33. log⁡(xn)=1n⋅log⁡(x)\log(\sqrt[n]{x}) = \frac{1}{n} \cdot \log(x) for any positive real number xx and positive integer nn.
  3. Apply property to a3a^3 and c3c^3: Apply the first property to the terms a3a^3 and c3c^3.log⁡(a3)=3×log⁡(a)=3×9=27\log(a^{3}) = 3 \times \log(a) = 3 \times 9 = 27log⁡(c3)=3×log⁡(c)=3×3=9\log(c^{3}) = 3 \times \log(c) = 3 \times 3 = 9
  4. Apply property to b23\sqrt[3]{b^2}: Apply the third property to the term b23\sqrt[3]{b^{2}}.log⁡(b23)=13⋅log⁡(b2)=13⋅2⋅log⁡(b)=23⋅12=8\log(\sqrt[3]{b^{2}}) = \frac{1}{3} \cdot \log(b^{2}) = \frac{1}{3} \cdot 2 \cdot \log(b) = \frac{2}{3} \cdot 12 = 8
  5. Combine logarithms: Apply the second property to combine the logarithms. \newlinelog⁡(a3c3b23)=log⁡(a3)+log⁡(c3)−log⁡(b23)\log\left(\frac{a^{3}c^{3}}{\sqrt[3]{b^{2}}}\right) = \log(a^{3}) + \log(c^{3}) - \log(\sqrt[3]{b^{2}})\newline=27+9−8= 27 + 9 - 8\newline=36−8= 36 - 8\newline=28= 28

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