Full solution

Q. Find the numerical value of the log expression.

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\begin{array}{c} \log a=10 \quad \log b=7 \quad \log c=-6 \\ \log \frac{\sqrt{c}}{a^{4} b^{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 $\log\left(\frac{\sqrt{c}}{a^{4}b^{5}}\right)$ . $\newline$ 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.
Expand Numerator and Denominator: Now, we apply the quotient rule to the given expression: $\log\left(\frac{\sqrt{c}}{a^{4}b^{5}}\right) = \log(\sqrt{c}) - \log(a^{4}b^{5})$ .
Simplify Using Rules: Next, we use the product rule of logarithms for the denominator, which states that $\log(a*b) = \log(a) + \log(b)$ , and the power rule, which states that $\log(a^n) = n\cdot\log(a)$ , to further expand the expression: $\newline$ $\log(\sqrt{c}) - (\log(a^{4}) + \log(b^{5}))$ .
Apply Power Rule: We can simplify the expression by applying the power rule to the terms $\log(a^{4})$ and $\log(b^{5})$ , and recognizing that $\sqrt{c}$ is $c^{\frac{1}{2}}$ : $\log(c^{\frac{1}{2}}) - (4\log(a) + 5\log(b))$ .
Substitute Given Values: Now we apply the power rule to $\log(c^{\frac{1}{2}})$ , which gives us $\frac{1}{2}\cdot\log(c)$ : $\frac{1}{2}\cdot\log(c) - (4\cdot\log(a) + 5\cdot\log(b))$ .
Perform Arithmetic Operations: We substitute the given values for $\log(a)$ , $\log(b)$ , and $\log(c)$ into the expression: $\newline$ $\left(\frac{1}{2}\right)*(-6) - (4*10 + 5*7)$ .
Combine Terms and Subtract: Perform the arithmetic operations: $-3 - (40 + 35)$ .
Get Numerical Value: Combine the terms inside the parentheses and then subtract from $-3$ : $-3 - 75$ .
Get Numerical Value: Combine the terms inside the parentheses and then subtract from $-3$ : $-3 - 75$ .Finally, we get the numerical value of the expression: $-3 - 75 = -78$ .