Answer
a) 35 years
b) 14.21 years
c) 7.27 years
Work Step by Step
We can solve this with an exponential equation. Recall that its general form is $ab^{x}$, where a is the principal balance, b is the common ratio, and x is the time in this scenario. Now, we can simply set the equation equal to 2, and a to 1 since we are finding the time necessary to double the price level.
a) $2 = 1.02^{x}$
Solving for x, we can take the natural logarithm of both sides, $\ln(2) = \ln(1.02^{x})$
Using logarithmic properties, we can bring the x into a multiple, so $\ln(2) = x \ln(1.02)$
Now solving for x, we get $x = \frac{\ln(2)}{\ln(1.02)} \approx$ 35 years
b) $2 = 1.05^{x}$
Repeat the process outlined in part (a), so $x = \frac{\ln(2)}{\ln(1.05)} \approx$ 14.21 years
c) $2 = 1.10^{x}$
Repeat the process outlined in part (a), so $x = \frac{\ln(2)}{\ln(1.10)} \approx$ 7.27 years
