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Find the numerical value of the log expression. {:[log a=-11quad log b=-5quad log c=6],[log ((a^(7)c^(5))/(b^(7)))]:} Answer:

Find the numerical value of the log expression.\newlinelog⁡a=−11log⁡b=−5log⁡c=6log⁡a7c5b7 \begin{array}{c} \log a=-11 \quad \log b=-5 \quad \log c=6 \\ \log \frac{a^{7} c^{5}}{b^{7}} \end{array} \newlineAnswer:

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Q. Find the numerical value of the log expression.\newlinelog⁡a=−11log⁡b=−5log⁡c=6log⁡a7c5b7 \begin{array}{c} \log a=-11 \quad \log b=-5 \quad \log c=6 \\ \log \frac{a^{7} c^{5}}{b^{7}} \end{array} \newlineAnswer:
  1. Given Logarithms: We are given the logarithms of aa, bb, and cc as log⁡a=−11\log a = -11, log⁡b=−5\log b = -5, and log⁡c=6\log c = 6. We need to find the value of log⁡(a7c5b7)\log\left(\frac{a^7c^5}{b^7}\right). We can use the properties of logarithms to simplify the expression.
  2. Simplify Expression: Using the properties of logarithms, we can break down the expression log⁡(a7c5b7)\log\left(\frac{a^7c^5}{b^7}\right) into the sum and difference of the individual logarithms: log⁡(a7)+log⁡(c5)−log⁡(b7)\log(a^7) + \log(c^5) - \log(b^7).
  3. Apply Power Rule: Now we apply the power rule of logarithms, which states that log⁡(xn)=nlog⁡(x)\log(x^n) = n\log(x), to each term: 7log⁡(a)+5log⁡(c)−7log⁡(b)7\log(a) + 5\log(c) - 7\log(b).
  4. Substitute Values: Substitute the given values of log⁡a\log a, log⁡b\log b, and log⁡c\log c into the expression: 7⋅(−11)+5⋅(6)−7⋅(−5)7\cdot(-11) + 5\cdot(6) - 7\cdot(-5).
  5. Perform Arithmetic Operations: Perform the arithmetic operations: −77+30+35-77 + 30 + 35.
  6. Combine Numbers: Combine the numbers to get the final result: −77+30+35=−47+35=−12-77 + 30 + 35 = -47 + 35 = -12.

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