Answer
$\underline{4^3=64}$
$log_4{64}=3$
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$log_4{2}=\frac{1}{2}$
$\underline{4^{\frac{1}{2}}=2}$
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$\underline{4^{\frac{3}{2}}=8}$
$\log_4{8}=\frac{3}{2}$
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$log_4{\frac{1}{16}}=-2$
$\underline{4^{-2}=\frac{1}{16}}$
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$log_4{\frac{1}{2}}=-\frac{1}{2}$
$\underline{4^{-\frac{1}{2}}=\frac{1}{2}}$
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$\underline{4^{-\frac{5}{2}}=\frac{1}{32}}$
$\log_4{\frac{1}{32}}=-\frac{5}{2}$
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.
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$4^3=64$
This is similar to: $log_4{64}=3$
$log_4{2}=\frac{1}{2}$
This one is same as: $4^{\frac{1}{2}}=2$
$4^{\frac{3}{2}}=8$
It's equal to: $\log_4{8}=\frac{3}{2}$
$log_4{\frac{1}{16}}=-2$
It's similar to: $4^{-2}=\frac{1}{16}$
$log_4{\frac{1}{2}}=-\frac{1}{2}$
It's similar to: $4^{-\frac{1}{2}}=\frac{1}{2}$
$4^{-\frac{5}{2}}=\frac{1}{32}$
It's similar to: $\log_4{\frac{1}{32}}=-\frac{5}{2}$