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Find the numerical value of the log expression. {:[log a=4quad log b=4quad log c=-4],[log ((a^(9)b^(9))/(c^(5)))]:}

Find the numerical value of the log expression.\newlinelog⁡a=4log⁡b=4log⁡c=−4log⁡a9b9c5 \begin{array}{c} \log a=4 \quad \log b=4 \quad \log c=-4 \\ \log \frac{a^{9} b^{9}}{c^{5}} \end{array}

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Quotient property of logarithmsright chevron icon

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Q. Find the numerical value of the log expression.\newlinelog⁡a=4log⁡b=4log⁡c=−4log⁡a9b9c5 \begin{array}{c} \log a=4 \quad \log b=4 \quad \log c=-4 \\ \log \frac{a^{9} b^{9}}{c^{5}} \end{array}
  1. Apply Power 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⁡(a9b9c5)\log\left(\frac{a^{9}b^{9}}{c^{5}}\right). To do this, we will use the properties of logarithms, specifically the power rule and the quotient rule.\newlinePower rule of logarithm: log⁡(an)=n⋅log⁡(a)\log(a^n) = n \cdot \log(a)\newlineQuotient rule of logarithm: log⁡(ab)=log⁡(a)−log⁡(b)\log\left(\frac{a}{b}\right) = \log(a) - \log(b)
  2. Substitute Given Values: First, let's apply the power rule to the expression log⁡(a9b9)/(c5)\log(a^{9}b^{9})/(c^{5}).log⁡(a9b9)/(c5)=log⁡(a9)+log⁡(b9)−log⁡(c5)\log(a^{9}b^{9})/(c^{5}) = \log(a^{9}) + \log(b^{9}) - \log(c^{5})Now we apply the power rule:log⁡(a9)=9⋅log⁡(a)\log(a^{9}) = 9 \cdot \log(a)log⁡(b9)=9⋅log⁡(b)\log(b^{9}) = 9 \cdot \log(b)log⁡(c5)=5⋅log⁡(c)\log(c^{5}) = 5 \cdot \log(c)
  3. Perform Arithmetic Operations: Next, we substitute the given values of log⁡a\log a, log⁡b\log b, and log⁡c\log c into the expression.log⁡(a9)+log⁡(b9)−log⁡(c5)=9×log⁡(a)+9×log⁡(b)−5×log⁡(c)=9×4+9×4−5×(−4)\log(a^{9}) + \log(b^{9}) - \log(c^{5}) = 9 \times \log(a) + 9 \times \log(b) - 5 \times \log(c) = 9 \times 4 + 9 \times 4 - 5 \times (-4)
  4. Perform Arithmetic Operations: Next, we substitute the given values of log⁡a\log a, log⁡b\log b, and log⁡c\log c into the expression.\newlinelog⁡(a9)+log⁡(b9)−log⁡(c5)=9⋅log⁡(a)+9⋅log⁡(b)−5⋅log⁡(c)\log(a^{9}) + \log(b^{9}) - \log(c^{5}) = 9 \cdot \log(a) + 9 \cdot \log(b) - 5 \cdot \log(c)\newline=9⋅4+9⋅4−5⋅(−4)= 9 \cdot 4 + 9 \cdot 4 - 5 \cdot (-4) Now we perform the arithmetic operations.\newline=36+36+20= 36 + 36 + 20\newline=72+20= 72 + 20\newline=92= 92

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