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

Identify Fourth Root of 512: Identify the value of the fourth root of 512. 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). 512 = 2^9, so 512^(1/4) = (2^9)^(1/4) = 2^(9/4) = 2^(2.25) = 2^2 * 2^(0.25) = 4 * 2^(0.25). 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 * 1.189207 = 4.756828 (approximately). 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 log base 8 of 4, which we can write as 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) = log_c(a) / log_c(b). So, log_(8)(4) = log_(2)(4) / log_(2)(8).Calculate Log Values: Calculate log base 2 of 4 and log base 2 of 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. Now we divide the results from Step 4 to find the value of log base 8 of 4. log_(8)(4) = log_(2)(4) / log_(2)(8) = 2 / 3.

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

Evaluate. $\newline$ $\log _{8} \sqrt[4]{512}$ $\newline$ Write your answer in simplest form.

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Precalculus

Power property of logarithms

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Q. Evaluate.

\newline

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

\newline

Write your answer in simplest form.

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 $\log_{8}(4)$ . $\newline$ $\log_{8}(4) = \frac{\log_{2}(4)}{\log_{2}(8)} = \frac{2}{3}.$