Is $(a/b)-1$ approximately equal to $log_e (a/b)$

I was reading an article where in one of the steps we were trying to calculate the daily return. It said

Return = (a / b) – 1


It then said, this equation can be approximated to:

Return = Log e (a/b)


Could someone explain a proof around how these are equal? Why $$log_e$$ (and not $$log$$ base of another value)?

Mathematics Asked by noi.m on November 21, 2021

Set $$x:=log_e (a/b)$$, $$a/b>0$$;

Then

$$(a/b) =e^x= 1+x+x^2/2! +...;$$

$$(a/b) =$$

$$1+log_e (a/b) + O((log_e (a/b))^2)$$.

Answered by Peter Szilas on November 21, 2021

Take it in the other direction $$log left(frac{a}{b}right)=log left(1+frac{a-b}{b}right)approx frac{a-b}{b}=frac{a}{b}-1$$

Answered by Claude Leibovici on November 21, 2021

This is explained by the Taylor's theorem, expanding to the first order:

$$log(1+x)approx log(1+x)_{x=0}+left(log(1+x)right)'_{x=0}x=frac x{1+0}=x.$$

For logarithms in other bases, it suffices to apply the conversion factor. (The natural logarithm is used because no factor is required by the derivative.)

With $$x:=dfrac ab-1$$,

$$logleft(frac abright)approx frac ab-1.$$

The closer to $$1$$ the ratio, the better the approximation.

In fact, you are replacing the curve by its tangent:

Answered by user65203 on November 21, 2021

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