The logarithm to base 10 of a number \(x\), written \(\log_{10}(x)\), is the exponent you raise 10 to get \(x\), so it is the number \(y\) that makes the equation \(10^y = x\) true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. \(\log_2(x)\) is the number \(y\) that makes the equation \(2^y = x\) true. The logarithm to the base \(e\) is called the natural logarithm, and is written \(\ln(x)\).

A logarithmic function is a constant multiple of a logarithm to some base, so it is a function given by \(f(x) = k \log_{a}(x)\) where \(k\) is any number and \(a\) is a positive number (10, 2, or \(e\) in this course). The graph of a typical logarithmic function is shown. Although the function grows very slowly, the graph does not have a horizontal asymptote.

The natural logarithm of \(x\), written \(\ln(x)\), is the log to the base \(e\) of \(x\). So it is the number \(y\) that makes the equation \(e^y = x\) true.