A logarithm is the opposite of an exponent. For example, $3^2=9$ and $\log_{3}(9)=2$. We use a logarithm with the base of the exponent we are solving for and put the output in the parenthesis.

$$a^b=c \to \log_{b}(c)=a$$

Logarithms give us the power to solve for exponents which would otherwise be quite a tedious task.

The Natural Logarithm¶

For query about the notation of the natural log see the “Popular Colloquialisms and Contradictions” section below.

The natural logarithm, or natural log, is simply the logarithm with the base of Euler’s constant, $e$.

$$\log_e(x)$$

Euler’s constant is a special number with numerous properties documented in its own article, but only relevant information will be duplicated here.

Popular Colloquialisms and Contradictions¶

The most commonly used logarithm could be said to be the natural logarithm for its wide variety of uses and natural tendency to appear around Euler’s constant.

Because of this, mathematicians will frequently omit writing the base of a logarithm and let the reader infer that it is meant to be a natural logarithm. However, this notation evolved differently in different geographical locations.

In the West, the common notation is to infer that $\log(x)$ is $\log_{10}(x)$ and use the alternate phrase $\ln(x)$ specifically for the natural log $\log_{e}(x)$. Everywhere else the standard notation is to use $\log(x)$ to mean $\log_{e}(x)$ and then write an explicit base if they mean to use any other base.

In effect many articles will either specifically express that $\log(x)$ is used as the natural log, or instead use $\ln$ and disregard the contradiction. Mathematicians are also generally flexible while reading to adapt to either notation. Regardless of which notation you choose to use, it is most important that your work is clearly legible and documented.