Definition of logarithm

The exponential $\exp_a(x) = a^x$ is a bijective function from $\mathbb{R}$ to $(0;+\infty)$. By the theorem on the existence of the inverse we can define its inverse function: this is precisely the logarithm.

Definition — Logarithm

Fixing a base $a > 0$, $a \ne 1$, the logarithm to base $a$ of a number $y > 0$ is the unique exponent to which $a$ must be raised in order to obtain $y$: $\log_a(y) = x \iff a^x = y.$ The logarithm function $\log_a:(0;+\infty)\to\mathbb{R}$ is the inverse of the exponential of base $a$.

Remark — How to read " $\log$"

$\log_2 8$ is read “logarithm to base $2$ of $8$” and equals $3$, because $2^3 = 8$. Similarly $\log_{10} 1000 = 3$, because $10^3 = 1000$, and $\log_{1/2} 8 = -3$, because $(1/2)^{-3} = 2^3 = 8$. The logarithm always answers the question “to what power?”.

History — Napier and Briggs

Logarithms were introduced at the start of the seventeenth century by the Scotsman John Napier (Italianised as Nepero) as a tool for turning multiplications into additions and simplifying astronomical calculations. Shortly afterwards the Englishman Henry Briggs proposed the use of base $10$, giving rise to the “decimal” logarithms that for centuries were tabulated and used with the slide rule.

Links

Topics: Logarithmic function
Concepts: Inverse function · Logarithm
Functions: Exponential function · Logarithmic function
People: Henry Briggs · John Napier (Nepero)