Question

evaluate the following expression without using a calculator. \\(\log_{8}(\sqrt{8})\\) \\(\log_{8}(\sqrt{8}) = \square\\)

Explanation:

Step1: Rewrite the square root as an exponent

Recall that $\sqrt{8} = 8^{\frac{1}{2}}$. So the expression becomes $\log_{8}(8^{\frac{1}{2}})$.

Step2: Apply the logarithm power rule

The power rule of logarithms states that $\log_{a}(x^{n}) = n\log_{a}(x)$. Here, $a = 8$, $x = 8$, and $n=\frac{1}{2}$. So we have $\frac{1}{2}\log_{8}(8)$.

Step3: Evaluate $\log_{8}(8)$

By the definition of a logarithm, $\log_{a}(a)=1$ for any positive $a
eq1$. So $\log_{8}(8) = 1$. Then $\frac{1}{2}\times1=\frac{1}{2}$.

Answer:

$\frac{1}{2}$