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Question

Mortgage formula I'm using:

$$M = P \left(i + \frac{i}{(1+i)^n-1}\right)$$

where $M =$ payment amount, $P =$ principle balance, $i =$ term interest rate, and $n =$ number of terms.

But now I'm trying to solve for $n$ and get stuck trying to fix the exponent. I think it has to do with logarithms and I guess I slept through those classes. The farthest I can get is:

$$(1+i)^n = \frac{i}{\frac{M}{P}-1}+1$$

Answer 1

If you take the natural logarithm (ln) of both sides, you get: $$n \ln (1+i) = \ln \left( \frac{i}{ \frac{M}{P}-1} + 1 \right) $$

This follows from the law $\ln(a^b) = b \ln(a)$. Then you are one step away from isolating $n$.

Answer 2

@mweiss pointed me in the right direction. (Thank you!!) For anyone who runs across this, here is the completed solution:

$$n = \frac{\ln\left( \frac{i}{\frac{M}{P}-i}+1\right)}{\ln\left(1+i\right)}$$