# Another Reason Natural Logarithms Are Natural

# Another Reason Natural Logarithms Are Natural

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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 inxcorresponds to an approximate 6% difference iny, 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:

10^{x} ≈ 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 log_{10} scale corresponds to an increase of about 2.3% on the original scale.

**Related post**: Approximation relating lg, ln, and log10

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