Answer
$f(x)=\log_{a}x$ is a function defined as the inverse of $g(x)=a^{x}.$
($\log_{a}x$ is the exponent needed to raise a in order to obtain x).
Exponential functions require a positive base in order to avoid complex (nonreal) function values.
For example, if the base of an exponential function was $a=-4,$ then $g(\displaystyle \frac{1}{2})=\sqrt{-4}=2i \notin R.$
This would lead to a "patchy" domain with a lot of numbers excluded... and difficult to study.
Being directly connected to exponential functions, the same restriction on the base is imposed on logarithmic functions.
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