The common logarithm of a number x is the exponent to which we must raise 10 to obtain x.
In other words, if 10y=x, then we say that y is the common logarithm of x,
and we can write y=log10(x). We often abbreviate log10(x) to just log(x).
Because 10y 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 −∞ as x approaches 0, and it goes off to ∞ (very, very slowly) as x tends
to ∞.
The logarithm has a number of properties that follow from its definition as an exponent:
log10(x⋅y)=log10(x)+log10(y);
log10(1y)=−log10(y);
log10(xy)=log10(x)−log10(y) (follows from (1) and (2));
log10((x)n)=n⋅log10(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 loge(x) or ln(x),
is the exponent to which we must raise e to obtain x (e=2.7182818284…).
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.