Find the numerical value of the log expression. {:[log a=-6quad log b=-9quad log c=-12],[log ((root(3)(b^(4)c^(7)))/(a^(4)))]:} Answer:

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Find the numerical value of the log expression.  {:[log a=-6quad log b=-9quad log c=-12],[log ((root(3)(b^(4)c^(7)))/(a^(4)))]:} Answer:

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Understand Given Values: Understand the given logarithmic values and the expression to evaluate. We are given: log a = -6 log b = -9 log c = -12 We need to find the value of: log ((root(3)(b^(4)c^(7)))/(a^(4))) Let's start by simplifying the expression using logarithmic rules.Apply Quotient Rule: Apply the quotient rule of logarithms to the expression. The quotient rule states that log(x/y) = log(x) - log(y). So, log ((root(3)(b^(4)c^(7)))/(a^(4))) = log (root(3)(b^(4)c^(7))) - log (a^(4))Apply Power Rule: Apply the power rule of logarithms to the expression. The power rule states that log(x^(n)) = n * log(x). So, log (a^(4)) = 4 * log(a) Since we know log(a) = -6, we can substitute to get: log (a^(4)) = 4 * (-6) = -24Apply Cube Root: Apply the cube root as a power of 1/3 to the logarithm. The cube root of a number can be expressed as the number raised to the power of 1/3. So, log (root(3)(b^(4)c^(7))) = log ((b^(4)c^(7))^(1/3)) Using the power rule again, we get: log ((b^(4)c^(7))^(1/3)) = (1/3) * log (b^(4)c^(7))Apply Product Rule: Apply the product rule of logarithms to the expression. The product rule states that log(xy) = log(x) + log(y). So, (1/3) * log (b^(4)c^(7)) = (1/3) * (log (b^(4)) + log (c^(7)))Apply Power Rule: Apply the power rule to the individual logarithms. Using the power rule again: log (b^(4)) = 4 * log(b) log (c^(7)) = 7 * log(c) Since we know log(b) = -9 and log(c) = -12, we can substitute to get: log (b^(4)) = 4 * (-9) = -36 log (c^(7)) = 7 * (-12) = -84Substitute Values: Substitute the values back into the expression. Now we have: (1/3) * (log (b^(4)) + log (c^(7))) = (1/3) * (-36 + (-84)) = (1/3) * (-120) = -40Combine Results: Combine the results to find the final value of the original expression. We have: log ((root(3)(b^(4)c^(7)))/(a^(4))) = log (root(3)(b^(4)c^(7))) - log (a^(4)) = -40 - (-24) = -40 + 24 = -16

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=-6 \quad \log b=-9 \quad \log c=-12 \\ \log \frac{\sqrt[3]{b^{4} c^{7}}}{a^{4}} \end{array}$ $\newline$ Answer:

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Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=-6 \quad \log b=-9 \quad \log c=-12 \\ \log \frac{\sqrt[3]{b^{4} c^{7}}}{a^{4}} \end{array}$ $\newline$ Answer: