Application of Logarithms in real world applications i.e. in finance.

Question

I need your help regarding real life example of $\log(2)$, I mean is it relevant to any concepts - may be in finance. Actually, I looked up on Google and here at this forum too. I can find things related to natural logarithms but nothing in particular related to $\log(2)$, could somebody please guide me on this? Thanks in advance.

Answer 1

The formula for the doubling time with a constant growth rate uses the natural logarithm of $2$, where $t$ is time (years, months, days, etc.), and $r$ is the growth rate expressed as $r \%$.

$$\text{Time to double} = t \frac{\ln 2}{\ln 1 + \frac{r}{100}}$$

This has many real-world applications, such as the 'Rule of $70$' (or $72$). Taking the first-order approximation of the Taylor series of $\ln(1+x)$, we have:

$$t \frac{\ln 2}{\ln 1 + \frac{r}{100}} \approx t \frac{\ln 2}{\frac{r}{100}} \approx t \frac{100 \ln 2}{r} \approx t\frac{70}{r}$$

The annual rate of return is equal to $\frac{r}{t}$ where $t$ is in years, which gives the more commonly known version of the rule of $70$:

$$\text{Time to double} \approx \frac{70}{\text{rate of return}}$$