What is the difference between common and natural logarithms?

1 Answer
George C.
Aug 16, 2015

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#

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