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$Find the numerical value of the log expression. {:[log a=7quad log b=4quad log c=6],[log ((b^(8))/(a^(6)c^(5)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=7 \quad \log b=4 \quad \log c=6 \\ \log \frac{b^{8}}{a^{6} c^{5}} \end{array}$ $\newline$ 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=7 \quad \log b=4 \quad \log c=6 \\ \log \frac{b^{8}}{a^{6} c^{5}} \end{array}

\newline

Answer:

Apply Quotient Rule: Using the given values for $\log a$ , $\log b$ , and $\log c$ , we need to evaluate the expression $\log\left(\frac{b^8}{a^6c^5}\right)$ . We can use the properties of logarithms to simplify the expression. $\newline$ First, apply the quotient rule of logarithms, which states that $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$ . $\newline$ \log\left(\frac{b^\(8\)}{a^\(6\)c^\(5\)}\right) = \log(b^\(8) - \log(a^ $6$ c^ $5$ )
Apply Product Rule: Next, apply the product rule of logarithms to the term $\log(a^6c^5)$ , which states that $\log(ab) = \log(a) + \log(b)$ . $\newline$ $\log(a^6c^5) = \log(a^6) + \log(c^5)$
Apply Power Rule: Now, apply the power rule of logarithms to each term, which states that $\log(a^n) = n\log(a)$ . $\newline$ $\log(b^8) = 8\log(b)$ $\newline$ $\log(a^6) = 6\log(a)$ $\newline$ $\log(c^5) = 5\log(c)$
Substitute Given Values: Substitute the given values for $\log a$ , $\log b$ , and $\log c$ into the expression. $\newline$ $\log(b^8) - (\log(a^6) + \log(c^5)) = 8\cdot\log(b) - (6\cdot\log(a) + 5\cdot\log(c))$ $\newline$ $= 8\cdot4 - (6\cdot7 + 5\cdot6)$
Perform Arithmetic Operations: Perform the arithmetic operations. $\newline$ $= 32 - (42 + 30)$ $\newline$ $= 32 - 72$ $\newline$ $= -40$

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