Final answer:
The formula solved for P gives the principal amount, whereas solving for n requires using a logarithmic function. The monthly payment formula includes compound interest, which is pivotal for loan amortization. To optimize loan repayment, making an equivalent of 13 monthly payments a year saves time and interest.
Step-by-step explanation:
Understanding the Monthly Payment and Amortization Formulas
When you solve for P in the monthly payment formula M = Pr(1+r)^(n) divided by ((1+r)^(n) - 1), you obtain the formula for the principal amount of the loan, given the monthly payment, interest rate, and number of payments. This would be the loan amortization formula. When solving for n, the number of payments, you would need to use the logarithmic function due to the exponent in the equation. Here's the catch, you can't just simply isolate n through algebraic means due to its placement in the exponent. This is where the logarithm becomes necessary. The monthly payment formula indeed takes compound interest into account, which is essential for amortizing loans, like a mortgage or car payment.
To find the monthly payments for a $300,000 loan with a 6% interest rate convertible monthly over 30 years, you would substitute these values into the monthly payment formula. If you increase the payments by a fraction of 12, effectively making 13 payments a year, you would save on total interest paid over the life of the loan and also reduce the time it takes to pay off the loan.