Exponential Form of a Logarithm The exponential form of a logarithm is the exponential equality the corresponds to a logarithmic equality.
For Example: The logarithmic equality $\log _4(256) = x$ has, as an exponential form, $4^x = 256$.
In general: For a logarithmic equality $\log _b(c) = a$, the exponential form is $b^a = c$.

Base (of a logarithm The base of a logarithm is the value of the base of the exponential that the logarithm is the inverse of.
For Example: Log with a base of $4$ is written $\log _4$ and is the function that represents the inverse of the exponential with the base 4, eg $4^x$. Thus $\log _4(4^x) = x$.

Argument (of a log) The argument of a log is the contents inside the log, which is the value of the exponential that you are trying to invert.
For Example: The expression $\log _4(256)$ has an argument of $256$.