Full solution

Q. Find the numerical value of the log expression.

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\begin{array}{c} \log a=2 \quad \log b=5 \quad \log c=-8 \\ \log \frac{b^{5}}{\sqrt{a^{3} c^{5}}} \end{array}

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Answer:

Apply Quotient Rule: Let's start by expressing the given logarithm using the properties of logarithms. $\newline$ We have the expression $\log\left(\frac{b^{5}}{\sqrt{a^{3}c^{5}}}\right)$ . $\newline$ Using the quotient rule of logarithms, which states that $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ , we can separate the numerator and the denominator.
Convert Square Root to Exponent: Now, we focus on the denominator, which contains a square root. The square root can be expressed as an exponent of $\frac{1}{2}$ . So, $\sqrt{a^{3}c^{5}}$ can be written as $(a^{3}c^{5})^{\frac{1}{2}}$ . Using the power rule of logarithms, which states that $\log(a^{n}) = n\log(a)$ , we can take the exponent outside the logarithm.
Apply Power Rule: Applying the power rule to both the numerator and the denominator, we get: $\log(b^{5}) - \log((a^{3}c^{5})^{\frac{1}{2}})$ . This simplifies to $5\log(b) - \left(\frac{1}{2}\right)\log(a^{3}c^{5})$ .
Apply Product Rule: Next, we apply the product rule of logarithms, which states that $\log(a \cdot b) = \log(a) + \log(b)$ , to the term $\log(a^{3}c^{5})$ . This gives us: $\newline$ $(\frac{1}{2})\cdot(\log(a^{3}) + \log(c^{5}))$ .
Apply Power Rule Again: Now, we apply the power rule again to the terms $\log(a^{3})$ and $\log(c^{5})$ , which gives us: $(\frac{1}{2})\cdot(3\cdot\log(a) + 5\cdot\log(c))$ .
Substitute Given Values: Substituting the given values of $\log a$ , $\log b$ , and $\log c$ into the expression, we get: $\newline$ $5\cdot\log(b) - \left(\frac{1}{2}\right)\cdot\left(3\cdot\log(a) + 5\cdot\log(c)\right)$ becomes $5\cdot5 - \left(\frac{1}{2}\right)\cdot\left(3\cdot2 + 5\cdot(-8)\right)$ .
Perform Arithmetic Operations: Performing the arithmetic operations, we have: $25 - (\frac{1}{2})*(6 - 40)$ which simplifies to $25 - (\frac{1}{2})*(-34)$ .
Calculate Final Value: Finally, we calculate the value of the expression: $25 - (\frac{1}{2})*(-34) = 25 - (-17) = 25 + 17 = 42$ .