Answer
$$\log_8{8}=1$$ $$\underline{8^1=8}$$
--
$$log_8{64}=2$$ $$\underline{8^2=64}$$
--
$$\underline{8^{\frac{2}{3}}=4}$$ $$log_8{4}=\frac{2}{3}$$
--
$$\underline{8^3=512}$$ $$log_8{512}=3$$
--
$$log_8{\frac{1}{8}}=-1$$ $$\underline{8^{-1}=\frac{1}{8}}$$
--
$$\underline{8^{-2}=\frac{1}{64}}$$ $$log_8{\frac{1}{64}}=-2$$
Work Step by Step
Before we begin, let's shortly overview what is logarithmic expression. $log_{a}{x}=b$ this is a logarithmic form which by exponential form means $a^b=x$. According to this we can easily fill the table.
$\log_8{8}=1$
This simply means: $8^1=8$
$log_8{64}=2$
It means: $8^2=64$
This time we have vice versa: $8^{\frac{2}{3}}=4$
And it means: $log_8{4}=\frac{2}{3}$
$8^3=512$
It stands for: $log_8{512}=3$
$log_8{\frac{1}{8}}=-1$
In this case: $8^{-1}=\frac{1}{8}$
And the last one: $8^{-2}=\frac{1}{64}$
It is similar as: $log_8{\frac{1}{64}}=-2$