# Is there a difference between using ln() and log() for calculating the periodic return of an asset

I am a highschool student with a small but decent knowledge of stocks. For a maths project, I'm investigating the probabilities and correlations of stocks.

I came across this formula for calculating the periodic daily return as a tool to help calculate historical volatility of a stock:

$$\text{Periodic Daily Return} = \ln(\frac{\text{Today's Stock Price}}{\text{Yesterday's Stock Price}})$$

I have a basic understanding of $$e$$ (in that it is the continuous compounding interest of $$\1$$ for one period of time at a rate of $$100\%$$), and hence I understand how e can be manipulated to get the above formula, and how it has advantages over a simple interest calculation.

I'm curious as to whether $$\log_{10}$$ or even $$\log_b$$ can be used to the same effect, and if not, why $$\ln$$ is more advantageous.

Any help would be much appreciated, cheers!

• I think the equation is $P(t)=P(0)\cdot e^{\delta \cdot t}$, For $t=1$ we get the daily return. Here we can take the logarithmus naturalis only. $$\delta=\ln\left(\frac{P(1)}{P(0)}\right)$$ – callculus May 18 at 23:43

It's not a big difference as $$\log x = \ln x / \ln 10$$ - changing logarithm base is just multiplication of it's value by constant. In terms of percents, $$\ln 10$$ is period you need to get tenfold increase at continuous 100% rate (and $$\ln e = 1$$ is period you need to get an increase in $$e$$ times).
You can use $$\log$$, but then you will need to divide result by $$\ln 10$$ if you want to use it instead of $$\ln$$. Depending on further formulas you will use the return in, it can make them a bit simpler or a bit more complex.
$$e$$ is good base for logarithm for several reasons - for example derivative $$\ln'(x) = \frac{1}{x}$$, and any other base will require to replace $$1$$ in numerator with something else.
• I can't imagine anything but convenience for using function $f(x)$ instead of $a\cdot f(x)$ - we can always divide by $a$ all occurences of $f$. – mihaild May 21 at 8:58