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log_(7)64

$8$ . $\log _{7} 64$

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Evaluate logarithms

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Q.

8

.

\log _{7} 64

Identify Base and Number: Identify the base of the logarithm and the number whose logarithm is to be found. $\newline$ In $\log_{7} 64$ , $7$ is the base and $64$ is the number. $\newline$ We need to express $64$ as a power of $7$ .
Express as Power of Base: Since $64$ is not an integer power of $7$ , we need to find a way to express it as a power of $7$ . We can use the change of base formula for logarithms: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ , where $c$ is a new base we choose. Let's choose the natural logarithm (base $e$ ) for this purpose. So, $\log_7(64) = \frac{\ln(64)}{\ln(7)}$ .
Use Change of Base Formula: Calculate the natural logarithms of $64$ and $7$ using a calculator. $\newline$ $\ln(64) \approx 4.15888$ $\newline$ $\ln(7) \approx 1.94591$
Calculate Natural Logarithms: Divide the natural logarithm of $64$ by the natural logarithm of $7$ to find the value of $\log_7(64)$ . $\log_7(64) \approx \frac{4.15888}{1.94591} \approx 2.138$ Since we cannot express $64$ as an exact power of $7$ , the result will be an approximation.

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