# What is a common logarithm or common log?

Aug 5, 2018

The inverse of the function $f \left(x\right) = {10}^{x}$

#### Explanation:

The function:

$f \left(x\right) = {10}^{x}$

is a continuous, monotonically increasing function from $\left(- \infty , \infty\right)$ onto $\left(0 , \infty\right)$

graph{10^x [-2.664, 2.338, -2, 12.16]}

Its inverse is the common logarithm:

${f}^{- 1} \left(y\right) = {\log}_{10} \left(y\right)$

which as a result is a continuous, monotonically increasing function from $\left(0 , \infty\right)$ onto $\left(- \infty , \infty\right)$.

graph{log x [-1, 12.203, -1.3, 1.3]}

Note that the exponential function satisfies:

${10}^{a} \cdot {10}^{b} = {10}^{a + b}$

Hence its inverse, the common logarithm satisfies:

${\log}_{10} x y = {\log}_{10} x + {\log}_{10} y$

Aug 5, 2018

As detailed below.

#### Explanation:

common logarithm is the logarithm with base 10.

It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as "standard logarithm".

Historically, it was known as logarithmus decimalis or logarithmus decadis. It is indicated by ${\log}_{10} \left(x\right)$.

On calculators, it is usually "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log".

To mitigate this ambiguity, the ISO 80000 specification recommends that ${\log}_{10} \left(x\right)$ should be written lg (x) and ${\log}_{e} \left(x\right)$ should be ln (x).

https://en.m.wikipedia.org/wiki/Common_logarithm