Question

use the elementary properties of logarithms to solve the following equation. write your answer as a fraction reduced to lowest terms. \\(\log_{128}(x) = \frac{3}{7}\\) answer how to enter your answer (opens in new window)

Explanation:

Step1: Recall logarithm definition

The logarithmic equation $\log_{b}(a) = c$ is equivalent to $b^{c}=a$. Here, $b = 128$, $c=\frac{3}{7}$, so we rewrite the equation as $x = 128^{\frac{3}{7}}$.

Step2: Simplify the base

Note that $128 = 2^{7}$. Substitute this into the equation: $x=(2^{7})^{\frac{3}{7}}$.

Step3: Apply exponent rule

Using the exponent rule $(a^{m})^{n}=a^{m\times n}$, we get $x = 2^{7\times\frac{3}{7}}$.

Step4: Simplify the exponent

Simplify $7\times\frac{3}{7}=3$, so $x = 2^{3}$.

Step5: Calculate the result

$2^{3}=8$.

Answer:

$8$