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Another Reason Natural Logarithms Are Natural

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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 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.

Related post: Approximation relating lg, ln, and log10

Hortonworks Sandbox for HDP and HDF is your chance to get started on learning, developing, testing and trying out new features. Each download comes preconfigured with interactive tutorials, sample data and developments from the Apache community.

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