Geometric definition of the natural logarithm

Parametric equations

The natural logarithm can be defined by $\ln t={\int }_1^t\frac{1}{x}dx$. When $t>1$, this is the positive area bounded by the hyperbola $y=1/x$, the $x$-axis, $x=1$, and $x=t$. When $0<t<1$, it is a signed area. The number $e$ satisfies $\ln e=1$. It is the unique positive number that makes this integral equal to $1$. $\ln$ and $e^x$ are inverses of each other.

The derivative relation is $\frac{d}{dt}\ln t=\frac{1}{t}$.

Interaction

• Area: drag Endpoint t and watch ${\int }_1^{t}1/xdx$ fill in, with the value shown as it changes
• e: the vertical line marks $t=e\approx 2.71828$, where that area is $1$
• Riemann rectangles: turn on a subdivision of the same area, then move Partitions n. The idea is the one on the Riemann sums page
• Inverse: switch to the exponential curve and the logarithm as inverses, symmetric across $y=x$

Related

• Logarithmic scale
• Exponential growth and decay
• Riemann sums

Further reading

• Natural logarithm (Wikipedia)
• e (mathematical constant) (Wikipedia)