Logarithms were very useful when they were first discovered because they greatly simplified complex calculations. Nowadays, we have calculators and computers that do all the work for us, but logarithms, in particular the natural logarithm, are still very important and useful today.
One of the first applications for the natural logarithm was to find coordinates on a map that uses the Mercator Projection . The Mercator Projection is very useful for navigation, but it is difficult to determine exact coordinates. The calculations were complicated and not well understood. Henry Bond was the first to notice that there might be a connection to the natural logarithm, but he was unable to prove it. James Gregory developed the first proof that ln could be used to determine coordinates on a Mercator Projection and Isaac Barrow developed a simpler proof shortly thereafter.
Spectroscopy deals with using a spectrometer to observe the behavior of particles and wavelengths. One of the spectra that is observed is the light spectrum. The natural log is used to determine an objects absorption of light. If I is the intensity of the light and I0 is the initial intensity then the absorbency is given by -ln(I0/I).
Since ln(x) and ex are inverses then we can use ln(ex) to solve for x in an equation involving ex. Some instances where we may use this would be finding the time t in a compound interest problem or a population growth problem. This process can also be used in carbon dating and other problems involving radioactive decay.
The natural log can also be used in statistics to transform data into a more linear form.