1958 AHSME Problems/Problem 12

Problem

If $P = \frac{s}{(1 + k)^n}$ then $n$ equals:

$\textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 + k)}}\qquad \textbf{(B)}\ \log{\left(\frac{s}{P(1 + k)}\right)}\qquad \textbf{(C)}\ \log{\left(\frac{s - P}{1 + k}\right)}\qquad \\ \textbf{(D)}\ \log{\left(\frac{s}{P}\right)} + \log{(1 + k)}\qquad \textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 + k))}}$

Solution

$P=\frac{s}{(1+k)^n}$

$(1+k)^n=\frac{s}{P}$

Take the $\log$ of each side.

$n \log(1+k) = \log\left(\frac{s}{P}\right)$

$n = \frac{\log\left(\frac{s}{P}\right)}{\log(1+k)} \to \boxed{\text{(A)}}$

1958 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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1958 AHSME Problems/Problem 12

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