# Another reason natural logarithms are natural

In mathematics, log means natural logarithm by default; the burden of explanation is on anyone taking logarithms to a different base. I elaborate on this a little here.

Looking through Andrew Gelman and Jennifer Hill’s regression book, I noticed a justification for natural logarithms I hadn’t thought about before.

We prefer natural logs (that is, logarithms base e) because, as described above, coefficients on the natural-log scale are directly interpretable as approximate proportional differences: with a coefficient of 0.06, a difference of 1 in x corresponds to an approximate 6% difference in y, and so forth.

This is because

exp(x) ≈ 1 + x

for small values of x based on a Taylor series expansion. So in Gelman and Hill’s example, a difference of 0.06 on a natural log scale corresponds to roughly multiplying by 1.06 on the original scale, i.e. a 6% increase.

The Taylor series expansion for exponents of 10 is not so tidy:

10x ≈ 1 + 2.302585 x

where 2.302585 is the numerical value of the natural log of 10. This means that a change of 0.01 on a log10 scale corresponds to an increase of about 2.3% on the original scale.

## 4 thoughts on “Another reason natural logarithms are natural”

1. Are you sure though that this is “another” reason? It seems to me this is a direct consequence of the first.

2. Barry Murphy

Similarly the growth rate of a time-series is approximately the natural log difference, often used as a data definition. Also the standard deviation of the log approximates (depending on skewness/kurtosis) the coefficient of variation. Further, changing the logarithmic base of regression variables changes only the constant. It would be nice to see one of your excellent posts on these.

3. hobohak

Is this not the actual reason according to Napier?