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$Find the numerical value of the log expression. {:[log a=2quad log b=2quad log c=2],[log ((c^(7))/(a^(9)b^(5)))]:} Answer:$

Find the numerical value of the log expression. $\newline$ $\begin{array}{c} \log a=2 \quad \log b=2 \quad \log c=2 \\ \log \frac{c^{7}}{a^{9} b^{5}} \end{array}$ $\newline$ Answer:

Full solution

Q. Find the numerical value of the log expression.

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\begin{array}{c} \log a=2 \quad \log b=2 \quad \log c=2 \\ \log \frac{c^{7}}{a^{9} b^{5}} \end{array}

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Answer:

Apply Rules: Using the quotient and power rules of logarithms, we can expand the given expression. $\newline$ Quotient rule of logarithm: $\log(\frac{a}{b}) = \log(a) - \log(b)$ $\newline$ Power rule of logarithm: $\log(a^n) = n \cdot \log(a)$ $\newline$ Let's apply these rules to the given expression $\log\left(\frac{c^{7}}{a^{9}b^{5}}\right)$ .
Separate Numerator and Denominator: First, apply the quotient rule to separate the logarithm of the numerator and the denominator: $\newline$ $\log\left(\frac{c^{7}}{a^{9}b^{5}}\right) = \log(c^{7}) - \log(a^{9}b^{5})$
Apply Product Rule: Next, apply the product rule to the logarithm in the denominator: $\newline$ Product rule of logarithm: $\log(a \times b) = \log(a) + \log(b)$ $\newline$ $\log(a^{9}b^{5}) = \log(a^{9}) + \log(b^{5})$
Apply Power Rule: Now, apply the power rule to each logarithm: $\newline$ $\log(c^{7}) = 7 \cdot \log(c)$ $\newline$ $\log(a^{9}) = 9 \cdot \log(a)$ $\newline$ $\log(b^{5}) = 5 \cdot \log(b)$
Substitute Given Values: Substitute the given values for $\log a$ , $\log b$ , and $\log c$ into the expression: $\newline$ $7 \times \log(c) - (9 \times \log(a) + 5 \times \log(b))$ $\newline$ $7 \times 2 - (9 \times 2 + 5 \times 2)$
Perform Arithmetic Operations: Perform the arithmetic operations: $\newline$ $14 - (18 + 10)$ $\newline$ $14 - 28$
Calculate Final Result: Calculate the final result: $14 - 28 = -14$