The water pump goes out on your car and it’s a $450 repair bill that you weren’t counting on. You have to put this repair on a credit card that has an interest rate of 15%, you can only make $50 payments each month until it’s paid off. How long will it take you to pay off the credit card debt?

To calculate the number of months needed to pay off the 450 debt on a credit card with a 15% annual interest rate and 50 monthly payments, we will use the given formula:

t = \frac{\log\left(\frac{50}{50 - 0.15 \times \frac{450}{12}}\right)}{\log\left(1 + \frac{0.15}{12}\right)}

Now, let's simplify the formula by evaluating the expressions inside the logarithms:

t = \frac{\log\left(\frac{50}{50 - 0.15 \times 37.5}\right)}{\log\left(1 + \frac{0.15}{12}\right)}
t = \frac{\log\left(\frac{50}{50 - 5.625}\right)}{\log\left(1 + 0.0125\right)}
t = \frac{\log\left(\frac{50}{44.375}\right)}{\log(1.0125)}
t = \frac{\log(1.12476)}{\log(1.0125)}
t \approx \frac{0.05144}{0.00510}
t \approx 10

Therefore, the number of months needed to pay off the 450 debt on your credit card with 50 monthly payments and a 15% annual interest rate is \boxed{10} months.