I'm trying to figure out why when I do not have an 'a' value (Compound Frequency) I can no longer use logarithms to solve for 'n' (number of years). Let the formula look like this:
FV=PV(1+(i/100)/a)^na
So by using the next formula:
(log(FV/PV))/(log(1+(i/100)/a))=n
It only appears to work when a is equal to 1. When I change it to something like 52, I end up get the incorrect number. Try using these variables:
- FV = 122.12
- PV = 100
- i = 10
- a = 52
I know that n=2, but using the second formula I constantly get 104. Why is this happening? Apologies for poor formatting, I'm just in a rush.
If you take logs of $\frac{FV}{PV}=\left(1+\frac{i}{100\cdot a}\right)^{n\cdot a}$ you get
$$\log\left(\frac{FV}{PV} \right)=n\cdot a \cdot \log\left(1+\frac{i}{100\cdot a} \right),$$
which leads to
$$n=\frac{\log\left(\frac{FV}{PV} \right)}{a \cdot \log\left(1+\frac{i}{100\cdot a} \right)}$$
Numerical example:
$$n=\frac{\log\left(\frac{122.12}{100} \right)}{52 \cdot \log\left(1+\frac{10}{100\cdot 52} \right)}= 2.00...$$
It is comprehensible?
Update
That´s the formula for compound Interest for monthly (or other period) compounding interest $FV=PV\cdot \left(1+\frac{r}a \right)^{n\cdot a}$. Similar formula can be seen here. Here r is the yearly interest rate in decimal representation, e.g. $r=0.04$ for $r=4\%$. In your case it is $r=\frac{i}{100}$, with $i=4$. So you divide the interest rate $\frac{i}{100}$ by $a$: $$\frac{\frac{i}{100}}{a}.$$ That is a double fraction as $$\frac{\frac{m}{n}}{\frac{s}{t}}$$ In your example we have $m=i, n=100, s=a$ and $t=1$. We divide a fraction by another fraction by multiplying the first fraction by the reciprocal of the second fraction. Thus we get $$\frac{m}{n}\cdot \frac{t}{s}=\frac{i}{100}\cdot \frac{1}{a}=\frac{i}{a\cdot 100}$$