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Find the numerical value of the log expression. {:[log a=-10quad log b=11quad log c=4],[log ((c^(4))/(a^(3)b^(3)))]:} Answer:

Find the numerical value of the log expression.\newlinelog⁡a=−10log⁡b=11log⁡c=4log⁡c4a3b3 \begin{array}{c} \log a=-10 \quad \log b=11 \quad \log c=4 \\ \log \frac{c^{4}}{a^{3} b^{3}} \end{array} \newlineAnswer:

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Q. Find the numerical value of the log expression.\newlinelog⁡a=−10log⁡b=11log⁡c=4log⁡c4a3b3 \begin{array}{c} \log a=-10 \quad \log b=11 \quad \log c=4 \\ \log \frac{c^{4}}{a^{3} b^{3}} \end{array} \newlineAnswer:
  1. Apply Quotient Rule: We are given the values of log⁡a\log a, log⁡b\log b, and log⁡c\log c. We need to find the value of log⁡(c4a3b3)\log\left(\frac{c^{4}}{a^{3}b^{3}}\right). Using the quotient rule of logarithms, which states that log⁡(xy)=log⁡(x)−log⁡(y)\log\left(\frac{x}{y}\right) = \log(x) - \log(y), we can separate the logarithm of the fraction into the difference of the logarithms of the numerator and the denominator.
  2. Apply Power Rule: Now, we apply the power rule of logarithms, which states that log⁡(xn)=n⋅log⁡(x)\log(x^n) = n \cdot \log(x), to the numerator and denominator separately.\newlineFor the numerator, we have c4c^{4}, so we apply the power rule to get 4⋅log⁡(c)4 \cdot \log(c).\newlineFor the denominator, we have a3a^{3} and b3b^{3}, so we apply the power rule to get 3⋅log⁡(a)3 \cdot \log(a) and 3⋅log⁡(b)3 \cdot \log(b) respectively.
  3. Substitute Given Values: We can now substitute the given values of log⁡a\log a, log⁡b\log b, and log⁡c\log c into the expression.\newlineThis gives us 4×log⁡(c)−(3×log⁡(a)+3×log⁡(b))=4×4−(3×(−10)+3×11)4 \times \log(c) - (3 \times \log(a) + 3 \times \log(b)) = 4 \times 4 - (3 \times (-10) + 3 \times 11).
  4. Perform Arithmetic Operations: Performing the arithmetic operations, we get 16−(−30+33)=16−316 - (-30 + 33) = 16 - 3.
  5. Calculate Final Result: Finally, we calculate the result of the subtraction, which is 16−3=1316 - 3 = 13.

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