The Common Logarithmic Function

The function f (x) = log10x is called the common logarithmic function. The common log function is often written as f (x) = log x -- when log is written without a base, the base is assumed to be 10. The "log" button on most calculators means log10.

log1000 = 3.
log0.01 = - 2.
log45 1.653 (using a calculator).

Figure %: f (x) = log x

The Natural Logarithmic Function

The number 2.71828… occurs often in mathematics and nature. For this reason, it is given a special name: e. Like Π, e is a number that mathematicians use often, and it is an irrational number, so it does not repeat or terminate. e has been calculated to many decimal places, but it is often rounded to e 2.718281828.


The function f (x) = logex is called the natural logarithmic function. The natural log function is often abbreviated f (x) = ln x--this is the way it appears on most calculators. ln ex = x and eln x = x.

lne4 = 4.
ln = - 2.
ln62 4.1271.
ln230 5.4381.
ln0.04 - 3.2189.

Figure %: f (x) = ln x