1952 AHSME Problems/Problem 18

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Problem

$\log p+\log q=\log(p+q)$ only if:

$\textbf{(A) \ }p=q=\text{zero}  \qquad \textbf{(B) \ }p=\frac{q^2}{1-q} \qquad \textbf{(C) \ }p=q=1 \qquad$

$\textbf{(D) \ }p=\frac{q}{q-1} \qquad \textbf{(E) \ }p=\frac{q}{q+1}$

Solution

$\log p+\log q=\log(p+q)\implies \log pq=\log(p+q)\implies pq=p+q\implies \boxed{\textbf{(D)}\ p=\frac{q}{q-1}}$

See also

1952 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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