Question

What is the difference between common and natural logarithms?

Answer 1

The base.

Essentially the common log is `log_10`, the inverse of `10^x`, while the natural log is `log_e`, the inverse of `e^x`.

Explanation:

The common logarithm is useful for base `10` calculations, especially in conjunction with scientific notation.

The natural logarithm `log_e x = ln x` is used more in algebra and calculus. Its inverse `e^x` has nice properties like `d/(dx) e^x = e^x`, `e^(i theta) = cos theta + i sin theta`, etc.

Another frequently used base for logarithms is `2`. The binary logarithm `log_2` is often used in computer science.

It is easy to convert between different logarithmic bases using the change of base formula:

`log_a b = (log_c b) / (log_c a)`

For example `log_10 x = (ln x)/(ln 10)` and `ln x = ln 10 * log_10 x`