Let's define i = monthly interest. How to get monthly interest from yearly interest depends on how the yearly interest is given. If it's given as APR, then you can just divide the yearly interest by 12 (there is some complication as to whether there's compounding of interest within a month, but unless your rate is absurdly high, that should be small enough that it won't affect the result much). If it's given as APY, it's useful to define r = 1+i. The monthly r is the yearly r raised to the one-twelfth power. Let's also represent the size of the monthly check with m, the principal you currently have as P, and the principal you have n month from now as P_n. Then we have
P_(n+1) = P_n+iP_n-m = (1+i)P_n-m = rP_n-m
That is, the principal each month is the rate applied to the previous principal, minus the monthly payment.
Now, suppose we guess that P_n can be written as Ar^n+B for some A and B. (If you're wondering where that guess came from, it's just a matter of being familiar with this sort of math problem.)
P_(n+1) = Ar^(n+1)+B = rP_n-m = r(A^n+B)-m = Ar^(n+1)+rB-m
So
Ar^(n+1)+B = Ar^(n+1)+rB-m and hence
B = rB-m
B-rB = -m
(1-r)B = -m
B = -m/(1-r)
B = m/(r-1)
I defined r as being 1+i, so i = r-1, so B = m/i
Next, we can take P_0 = P = Ar^0+B = A+B = A + m/i
So A = P - m/i
This gives the formula
P_n = (P-m/i)r^n + m/i
You're looking for how look it takes to pay off the loan, so you want P_n = 0
(P-m/i)r^n + m/i = 0
m/i = (m/i-P)r^n
(m/i)/(m/i-P) = r^n
m/(m-Pi) = r^n
log_r[m/(m-Pi)] = n
So:
Take the current principal and multiply by the interest; hat gives you how much money is being added in interest each month. Subtract that number from the monthly payment; that gives you how much you're currently contributing to the principal. Divide the monthly payment by that number; that tell you what the ratio is between the total amount you're paying and how much you're paying is going towards the principal. Then take the log of that number, base r. Since calculators generally don't have arbitrary bases, you can also take the log of that number, any base, then divide by the log of r in that same base.
For example, suppose you have a yearly interest rate of 6% APR. That gives i = .5% and r = 1.005. Now suppose you have a balance of $100,000 and you're paying $1000 a month. So P = 100,000 and m = 200. You have .5%*$100,000= $500 of interest accruing this month. Your monthly payment is twice as large as that, so you need the log base r of 2. The natural log of 2 is .301, and the natural log of 1.005 is .0021. The ratio between those is .301/.0021 = 138.97, so you have 139 payments left (11 years and 7 months).
