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What is the value of
log_(8)root(4)(512) ?
Answer:

What is the value of $\log _{8} \sqrt[4]{512}$ ? $\newline$ Answer:

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Power property of logarithms

Full solution

Q. What is the value of

\log _{8} \sqrt[4]{512}

?

\newline

Answer:

Identify Fourth Root of $512$ : Identify the value of the fourth root of $512$ . $\newline$ The fourth root of $512$ is the number that, when raised to the power of $4$ , equals $512$ . We can calculate this as $512^{(1/4)}$ . $\newline$ $512 = 2^9$ , so $512^{(1/4)} = (2^9)^{(1/4)} = 2^{(9/4)} = 2^{2.25} = 2^2 \times 2^{0.25} = 4 \times 2^{0.25}$ . $\newline$ Since $2^{0.25}$ is the fourth root of $2$ , and the fourth root of $2$ is $2^{1/4} = 1.189207$ (approximately), we have $4 \times 1.189207 = 4.756828$ (approximately). $\newline$ However, we know that $2^2$ is exactly $4$ , so the fourth root of $512$ is exactly $2^2 = 4$ .
Apply Logarithm: Apply the logarithm to the fourth root of $512$ . We need to find $ext{log}_8(4)$ , which we can write as $ext{log}_{8}(4)$ .
Convert Base to $2$ : Convert the base of the logarithm from $8$ to $2$ . Since $8$ is $2^3$ , we can use the change of base formula to convert the base from $8$ to $2$ . The change of base formula is $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ . So, $\log_{8}(4) = \frac{\log_{2}(4)}{\log_{2}(8)}$ .
Calculate Log Values: Calculate $\log_{2} 4$ and $\log_{2} 8$ . We know that $4$ is $2^{2}$ , so $\log_{2}(4) = 2$ . We also know that $8$ is $2^{3}$ , so $\log_{2}(8) = 3$ .
Divide Logarithms: Divide the two logarithms to find the value. $\newline$ Now we divide the results from Step $4$ to find the value of $\log_{8}(4)$ . $\newline$ $\log_{8}(4) = \frac{\log_{2}(4)}{\log_{2}(8)} = \frac{2}{3}.$

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