Question

use the elementary properties of logarithms to solve the following equation. write your answer as a fraction reduced to lowest terms.
\log_{8}(x) = \frac{2}{3}
answer how to enter your answer (opens in new window)
3 points
x =

Explanation:

Step1: Recall logarithm definition

The logarithmic equation $\log_{b}(a) = c$ is equivalent to $b^{c}=a$ (by the definition of logarithms, where $b>0,b
eq1$). Here, $b = 8$, $c=\frac{2}{3}$, and we need to find $a=x$.

Step2: Apply the definition

Substitute into the exponential form: $x = 8^{\frac{2}{3}}$.

Step3: Simplify the exponent

We know that $8 = 2^{3}$, so $8^{\frac{2}{3}}=(2^{3})^{\frac{2}{3}}$. Using the power - of - a - power rule $(a^{m})^{n}=a^{mn}$, we get $2^{3\times\frac{2}{3}}=2^{2}$.

Step4: Calculate the result

$2^{2}=4$.

Answer:

$x = 4$