Question

question 8 of 24 step 1 of 1 use the elementary properties of logarithms to solve the following equation. write your answer as a fraction reduced to lowest terms. \\( \log_{16}(x) = \frac{3}{4} \\) answer how to enter your answer (opens in new window) 2 points keypad keyboard shortcuts

Explanation:

Step1: Recall logarithm definition

The logarithmic equation $\log_{b}(x) = y$ is equivalent to $x = b^{y}$ in exponential form. Here, $b = 16$, $y=\frac{3}{4}$, so we rewrite the equation as $x = 16^{\frac{3}{4}}$.

Step2: Simplify the exponent

We know that $16 = 2^{4}$, so substitute that in: $x=(2^{4})^{\frac{3}{4}}$. Using the power - of - a - power rule $(a^{m})^{n}=a^{m\times n}$, we get $x = 2^{4\times\frac{3}{4}}$.

Step3: Calculate the exponent

Simplify the exponent: $4\times\frac{3}{4}=3$. So $x = 2^{3}$.

Step4: Evaluate the power

Calculate $2^{3}=8$.

Answer:

8