The common logarithm of a number $x$ is the exponent to which we must raise 10 to obtain $x$.
In other words, if $10^y = x$, then we say that $y$ is the common logarithm of $x$,
and we can write $y = \log_{10}(x)$. We often abbreviate $\log_{10}(x)$ to just $\log(x)$.
Because $10^y$ is always positive, $\log(x)$ is only defined for positive $x$ values.
We can plot $\log(x)$ against $x$ to obtain the function graph shown below. This function
goes to $-\infty$ as $x$ approaches 0, and it goes off to $\infty$ (very, very slowly) as $x$ tends
to $\infty$.
The logarithm has a number of properties that follow from its definition as an exponent:
$\log_{10}(\frac{x}{y}) = \log_{10}(x) - \log_{10}(y)$ (follows from (1) and (2));
$\log_{10}((x)^n) = n \cdot \log_{10}(x)$
Other logarithm functions can be defined for other positive "bases". For example, the other most
common logarithm is the natural logarithm: the natural logarithm of $x$, written $\log_{e}(x)$ or $\ln(x)$,
is the exponent to which we must raise $e$ to obtain $x$ ($e = 2.7182818284\ldots$).
Any logarithm base will have a function graph having the same general shape as the common logarithm,
and any logarithm base will enjoy analogous versions of the properties detailed above.
