Question 308449
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I'll do you one better.  Rather than answering that exact question, I'll give you a formula calculating the time to make any desired multiplication of your principal at any annual interest rate.


The formula for Future Value, *[tex \Large A], of an investment, *[tex \Large P], at an annual interest rate of *[tex \Large r], for a period of  *[tex \Large t] years is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ A\ =\ Pe^{rt}]


Where *[tex \Large e] is the base of the natural logarithms.


For your specific problem, we want *[tex \Large A] to be equal to *[tex \Large 2P], which means that we want to determine the value of *[tex \Large t] that makes *[tex \Large e^{rt}\ =\ 2] when *[tex \Large r\ =\ 0.08].


So, in general, for a given multiplier, *[tex \Large m], we want *[tex \Large \frac{A}{P}\ =\ m].  (*[tex \Large m\ =\ 2] in your problem)


That means that we want:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ e^{rt}\ =\ m] 


Take the natural log of both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln(e^{rt})\ =\ \ln(m)]


Apply the rule for raising a the argument of a logarithm to a power:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ rt\ln(e)\ =\ \ln(m)]


Recall that *[tex \Large \ln(e)\ =\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ rt =\ \ln(m)]


And then divide by *[tex \Large r]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t =\ \frac{\ln(m)}{r}]


For this specific problem, plug in 0.08 for *[tex \Large r] and 2 for *[tex \Large m].  Then punch the buttons on your calculator.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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