Logarithms were first introduced by John Napier in the \(17^{\textrm{th}}\) century for the purpose of simplifying calculations.

This was accomplished with the development of logarithmic tables and, soon after, with logarithmic scales on a slide rule.

With the introduction of the scientific calculator in the mid-1970s, this application of logarithms for computations became somewhat obsolete; however, logarithms are still used today in many areas such as

• scientific formulas and scales (the pH scale in chemistry and the Richter scale for measuring and comparing the intensity of earthquakes),
• astronomy (order of magnitude calculations comparing relative size of massive bodies),
• modelling and solving problems involving exponential growth and decay, and
• many areas of calculus.

We will begin our study of logarithms by introducing and exploring the logarithmic function.

The logarithmic function is simply the inverse of the exponential function.

If\(\quad\)\(f(x) = c^x,~ c \gt 0,~ c \neq 1\),\(\quad\)then\(\quad f^{-1}(x) = \log_{c}(x), ~c \gt 0, ~c \neq 1 \).

Equivalent Forms

\(x = c^y \quad \Leftrightarrow \quad \log_{c}(x) = y\) where \( c > 0, ~c \neq 1\)

Similarly,

In general, let's consider

The base of the logarithm is \(\textcolor{BrickRed}{c}\).

The value of the logarithm, \(\textcolor{Mulberry}{n}\), is the exponent to which the base, \(\textcolor{BrickRed}{c}\), must be raised to yield the value of \(\textcolor{NavyBlue}{m}\).

Thus, \(\textcolor{NavyBlue}{m} = \textcolor{BrickRed}{c}^{\textcolor{Mulberry}{n}}\), so \(\textcolor{NavyBlue}{m}\) is the value of the power. \(\textcolor{NavyBlue}{m}\) is also referred to as the argument of the logarithm.