The Natural Logarithm

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The natural logarithm is the logarithm with base equal to e.

$\displaystyle \log_e (x) = \ln(x)$

The natural logarithm can be written as $\log_e x$ but is usually written as $\ln x$. The two letters l and n are reversed from the order in English because it arises from the French (logarithm naturalle). 

Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$, also arises in naturally in many contexts. It is used much more frequently in physics, chemistry, and higher mathematics than other logarithmic functions. For example, the doubling time for a population which is growing exponentially is usually given as ${\ln 2 \over k}$ where $k$ is the growth rate, and the half-life of a radioactive substance is usually given as ${\ln 2 \over \lambda}$ where $\lambda$ is the decay constant. 

Graphing $y=\ln(x)$

The function slowly grows to positive infinity as $x$ increases and rapidly goes to negative infinity as $x$ approaches $0$ ("slowly" and "rapidly" as compared to any power law of $x$). The $y$-axis is an asymptote. The graph of the natural logarithm lies between that of $y=\log_2 x$ and $y=\log_3 x$. Its value at $x=1$ is $0$, while its value at $x=e$ is $1$.

The graphs of $\log_2 x$, $\ln x$, and $\log_3 x$

The graph of the natural logarithm lies between the base 2 and the base 3 logarithms.

Solving Equations Using $\ln(x)$

The natural logarithm function can be used to solve equations in which the variable is in an exponent. 

Example: Find the positive root of the equation $3^{x^2-1}=8$ 

The first step is to take the natural logarithm of both sides:

$​\displaystyle \ln (3^{x^2-1}) = \ln 8$

Using the power rule of logarithms it can then be written as:

 $\displaystyle (x^2-1) \ln 3 = \ln 8$

Dividing both sides by $\ln(3)$ gives:

$\displaystyle x^2-1={\ln 8 \over \ln 3}$ 

Thus the positive solution is $x=\sqrt{{\ln 8 \over \ln 3} + 1}.$ This can be calculated (approximately) with any scientific handheld calculator.