Finding Curved Areas

Modern Thinking

The natural logarithm is the inverse of the exponential function.

e^log_ex= x =log_e(e^x)

But, the natural log was not always thought about or defined this way.

A Need for Calculus

When the natural log was introduced, before the beast we know as calculus arrived on the scene, we had no easy way to find the areas under curves. The 1/x curve was of specific interest.

This leads to the formal definition of the natural logarithm:

log_ex= integral[1..∞] 1/x dx

We knew that the integral above was a logarithm, because it behaves like a logarithm. It exhibits the property that log_e(ab)=log_ea + log_eb