Full solution

Q. Find the numerical value of the log expression.

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\begin{array}{cc} \log a=1 \quad & \log b=6 \quad \log c=9 \\ \log \frac{a^{4}}{\sqrt[3]{b^{7} c^{4}}} \end{array}

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Answer:

Identify Given Values: Identify the given logarithmic values and the expression to be evaluated. $\newline$ We are given $\log a = 1$ , $\log b = 6$ , and $\log c = 9$ . We need to find the value of $\log \left(\frac{a^{4}}{\sqrt[3]{b^{7}c^{4}}}\right)$ .
Apply Power Rule: Apply the logarithm power rule to the numerator of the expression. $\newline$ The power rule of logarithms states that $\log(a^n) = n \cdot \log(a)$ . Therefore, $\log(a^{4}) = 4 \cdot \log(a)$ . $\newline$ Since $\log a = 1$ , we have $\log(a^{4}) = 4 \cdot 1 = 4$ .
Apply Power Rule to Denominator: Apply the logarithm power rule to the denominator of the expression. $\newline$ We have a cube root in the denominator, which can be written as a power of $\frac{1}{3}$ . The expression inside the cube root is $b^{7}c^{4}$ , which can be separated using the product rule of logarithms: $\log(b^{7}c^{4}) = \log(b^{7}) + \log(c^{4})$ . $\newline$ Now apply the power rule: $\log(b^{7}) = 7 \cdot \log(b)$ and $\log(c^{4}) = 4 \cdot \log(c)$ . $\newline$ Since $\log b = 6$ and $\log c = 9$ , we have $\log(b^{7}) = 7 \cdot 6 = 42$ and $\log(c^{4}) = 4 \cdot 9 = 36$ .
Combine Denominator Logarithms: Combine the logarithms of the denominator and apply the cube root. $\newline$ We have $\log(b^{7}) + \log(c^{4}) = 42 + 36 = 78$ . $\newline$ Now, we need to apply the cube root, which is the same as multiplying by $1/3$ : $(1/3) \times \log(b^{7}c^{4}) = (1/3) \times 78 = 26$ .
Use Quotient Rule: Use the quotient rule of logarithms to combine the numerator and denominator. $\newline$ The quotient rule of logarithms states that $\log(\frac{a}{b}) = \log(a) - \log(b)$ . We have $\log(a^{4}) - \log(\sqrt[3]{b^{7}c^{4}}) = 4 - 26$ .
Calculate Final Value: Calculate the final numerical value. $\newline$ Subtract the value found for the denominator from the numerator: $4 - 26 = -22$ .