Introduction: A function of the form $y(x)=\;^a\!\log x, (x>0)$ where $a$ ($a\neq 1$) is a positive number is called a logarithmic function with base $a$.
Meaning of the logarithmic function with base $a$:
$y(x)=\;^a\!\log x$ means: to $x$ belongs the $y$ that satisfies $a^y=x$.
Example: For instance $y(x)=\;^2\!\log x$ gives:
$$\begin{align*}^2\!\log 1=0, &\text{ since }&2^0&=&1\end{align*}$$
$$\begin{align*}^2\!\log 2=1, &\text{ since }&2^1&=&2\end{align*}$$
$$\begin{align*}^2\!\log 8=3, &\text{ since }&2^3&=&8\end{align*}$$