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$Find the numerical value of the log expression. {:[log a=-3quad log b=-11quad log c=8],[log ((root(3)(a))/(b^(4)c^(5)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=-3 \quad \log b=-11 \quad \log c=8 \\ \log \frac{\sqrt[3]{a}}{b^{4} c^{5}} \end{array}$ $\newline$ Answer:

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Quotient property of logarithms

Full solution

Q. Find the numerical value of the log expression.

\newline

\begin{array}{c} \log a=-3 \quad \log b=-11 \quad \log c=8 \\ \log \frac{\sqrt[3]{a}}{b^{4} c^{5}} \end{array}

\newline

Answer:

Apply Properties of Logarithms: Let's start by applying the properties of logarithms to the expression $\log \left( \frac{\sqrt[3]{a}}{b^4 c^5} \right)$ . $\newline$ We can use the quotient rule of logarithms, which states that $\log \left( \frac{x}{y} \right) = \log(x) - \log(y)$ , and the power rule, which states that $\log(x^k) = k \cdot \log(x)$ .
Apply Quotient Rule: First, apply the quotient rule to the given expression: $\newline$ $\log \left( \frac{\sqrt[3]{a}}{b^4 c^5} \right) = \log(\sqrt[3]{a}) - \log(b^4 c^5)$ .
Apply Power Rule: Next, apply the power rule to the terms $\log(b^4)$ and $\log(c^5)$ : $\newline$ $\log(\sqrt[3]{a}) - \log(b^4 c^5) = \log(\sqrt[3]{a}) - (4 \cdot \log(b) + 5 \cdot \log(c))$ .
Express as Power of a: Now, we need to express $\log(\sqrt[3]{a})$ as a power of $a$ : $\newline$ Since $\sqrt[3]{a} = a^{1/3}$ , we can write $\log(\sqrt[3]{a})$ as $\frac{1}{3} \cdot \log(a)$ .
Substitute Given Values: Substitute the given values of $\log(a)$ , $\log(b)$ , and $\log(c)$ into the expression: $\newline$ $\frac{1}{3} \cdot \log(a) - (4 \cdot \log(b) + 5 \cdot \log(c)) = \frac{1}{3} \cdot (-3) - (4 \cdot (-11) + 5 \cdot 8)$ .
Perform Arithmetic Operations: Perform the arithmetic operations: $\newline$ $\frac{1}{3} \cdot (-3) - (4 \cdot (-11) + 5 \cdot 8) = -1 - (-44 + 40) = -1 - (-4) = -1 + 4 = 3$ .

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