1958 AHSME Problems/Problem 12

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Problem

If $P \equal{} \frac{s}{(1 \plus{} k)^n}$ (Error compiling LaTeX. Unknown error_msg) then $n$ equals:

$\textbf{(A)}\ \frac{\log{\left(\frac{s}{P}\right)}}{\log{(1 \plus{} k)}}\qquad \textbf{(B)}\ \log{\left(\frac{s}{P(1 \plus{} k)}\right)}\qquad \textbf{(C)}\ \log{\left(\frac{s \minus{} P}{1 \plus{} k}\right)}\qquad \ \textbf{(D)}\ \log{\left(\frac{s}{P}\right)} \plus{} \log{(1 \plus{} k)}\qquad \textbf{(E)}\ \frac{\log{(s)}}{\log{(P(1 \plus{} k))}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

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

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

Take the $\log$ of each side.

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

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


See also

1958 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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