The logarithm was first invented to perform hard calculations because of the identity \textstyle \log(ab)=\log(a)+\log(b) Since adding large numbers is far more convenient than multiplying them by hand, logarithms were a good way to calculate, or at worst approximate this large result. However, with the development of integral calculus, it was seen that the logarithm was much more vital. Euler's Mechanica talks about the Napier constant or Euler's number, e, which is defined as
e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}
or, equivalently,
e=\sum_{k=0}^{\infty}\frac{1}{k!}
The constant was first discovered by Bernoulli when studying compound interests. He used the first definition. Euler was the one to prove that the first definition is equivalent to the second, and the latter converges much more rapidly. Euler also showed that
e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}
If you differentiate term by term in this series, you will see that the derivative of this function, e to the x, is the function itself. Euler named it the exponential function and it is the name we use now.
The inverse of the exponential function is also important. Since exponential function satisfies the fundamental property of exponentiation, its inverse will be a logarithm, more precisely, the logarithm with base e. This logarithm is called the natural logarithm. This name follows from the simplicity of the definition of the natural logarithm: it is simply the area under the curve f(x)=1/x from 1 to n (if it was 0 to n, the area is infinite.) In other words, the area is the natural logarithm of n.
The natural logarithm is denoted in two ways in mathematics. One is ln(x), following from the French equivalent of the word "natural logarithm", which is "logarithm naturalis".
The other one is simply log(x), without any base notation.
Since logarithms satisfy another identity, \textstyle \log_{b}(a)=\log(a)/\log(b), it is possible to express any logarithm explicitly in terms of the natural logarithm.
All of what we wrote can be summed up in some equations:
1) The natural logarithm is the only solution to the equation (for f(x)):
\displaystyle \exp(f(x))=x
2) The natural logarithm can be defined as follows:
\log(n)=\int_{1}^{n}\frac{1}{x}dx
3) The exponential function satisfies
\frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x
From these, we can easily see that
4) The natural logarithm satisfies
\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\log(x)=\frac{1}{x}
The natural logarithm is involved in the solutions to many integrals. For example, we can solve the general integral
\displaystyle \int \frac{1}{ax+b}dx
using the natural logarithm, using the substitution u=ax+b, du=a dx:
\displaystyle \int \frac{1}{ax+b}dx=\displaystyle \frac{1}{a}\int \frac{1}{u}du=\frac{\log|u|}{a}+C=\frac{\log|ax+b|}{a}+C
The natural logarithm also arises in the concept of series. For example, consider the series
\displaystyle \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}
We can easily solve this by the Maclaurin series expansion of the natural logarithm:
\displaystyle \log(x)=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(x-1)^k}{k}
Substituting x=2 simply yields the result:
\displaystyle \log(2)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}
Yet another example of the natural logarithm can be the series
\displaystyle \lim_{n\to\infty}\sum^{2n}_{k=n}\frac{1}{k}
which is, again, the natural logarithm of two.
Logarithms are also used to describe a specific model of growth: one that has decaying rate of growth that approaches 0. Such a growth is called logarithmic growth. For example, the harmonic series grow logarithmically; because their growth at the nth term is 1/n, which approaches zero as n tends to infinity.
Logarithms are involved in many of physics equations, such as Newton's Law of Cooling, which can be expressed as
\displaystyle \frac{\mathrm{d}T}{\mathrm{d}t}=-k(T-T_a)
The solution of this differential equation for the time yields a logarithmic expression that involves the natural logarithm. It is worth noting that such a kind of growth (or decay) would grow (or decay) logarithmically.
Finally, I will conclude with some identities involving the natural logarithm and the exponential function.
\displaystyle e^{ix}=\cos(x) + i\sin(x)
\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{n} - \log(n)=\gamma
and is convergent.
\int \log(x)dx=x\log(x)-x+C
\sum_{p\,prime}\log\left(\frac{p}{p-1}\right)=\infty
e^{i\pi}=-1
