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.