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$Find the numerical value of the log expression. {:[log a=-6quad log b=10quad log c=7],[log ((sqrtb)/(a^(5)c^(7)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=-6 \quad \log b=10 \quad \log c=7 \\ \log \frac{\sqrt{b}}{a^{5} c^{7}} \end{array}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Find the numerical value of the log expression.

\newline

\begin{array}{c} \log a=-6 \quad \log b=10 \quad \log c=7 \\ \log \frac{\sqrt{b}}{a^{5} c^{7}} \end{array}

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Answer:

Apply Logarithm Rules: We are given the values of $\log a$ , $\log b$ , and $\log c$ , and we need to find the value of $\log\left(\frac{\sqrt{b}}{a^{5}c^{7}}\right)$ . First, let's apply the logarithm rules to simplify the expression. Using the quotient rule of logarithms, which states that $\log\left(\frac{x}{y}\right) = \log(x) - \log(y)$ , we can write: $\log\left(\frac{\sqrt{b}}{a^{5}c^{7}}\right) = \log(\sqrt{b}) - \log(a^{5}c^{7})$
Simplify $\log(\sqrt{b})$ : Now let's simplify $\log(\sqrt{b})$ . The square root can be expressed as a power of $\frac{1}{2}$ , so $\log(\sqrt{b}) = \log(b^{\frac{1}{2}})$ . Using the power rule of logarithms, which states that $\log(x^y) = y\cdot\log(x)$ , we get: $\log(b^{\frac{1}{2}}) = (\frac{1}{2})\cdot\log(b)$ Since we know $\log b = 10$ , we can substitute this value in: (\frac{$1$}{$2$})\cdot\log(b) = (\frac{$1$}{$2$})\cdot\(10 = $5$
Simplify $\log(a^{5}c^{7})$ : Next, we need to simplify $\log(a^{5}c^{7})$ . Using the product rule of logarithms, which states that $\log(xy) = \log(x) + \log(y)$ , we can write: $\log(a^{5}c^{7}) = \log(a^{5}) + \log(c^{7})$ Now we apply the power rule to both terms: $\log(a^{5}) + \log(c^{7}) = 5\cdot\log(a) + 7\cdot\log(c)$ Substituting the given values for $\log a$ and $\log c$ : $5\cdot\log(a) + 7\cdot\log(c) = 5\cdot(-6) + 7\cdot7 = -30 + 49 = 19$
Perform Subtraction: Now we have the values for both parts of our original expression: $\newline$ $\log\left(\frac{\sqrt{b}}{a^{5}c^{7}}\right) = \log(\sqrt{b}) - \log(a^{5}c^{7}) = 5 - 19$ $\newline$ We can now perform the subtraction: $\newline$ $5 - 19 = -14$