1963 AHSME Problems/Problem 5

Problem

If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, then:

$\textbf{(A)}\ x<0 \qquad \textbf{(B)}\ -1<x<1 \qquad \textbf{(C)}\ 0<x\le 1 \\ \textbf{(D)}\ -1<x<0 \qquad \textbf{(E)}\ 0<x<1$

Solution

Let $\log_{10} x = a$, so $10^a = x$. Note that $10^0 = 1$ and that $10^a$ increases as $a$ increases. Since $a < 0$, $10^a < 1$. However, for all $a$, $10^a > 0$, so $0 < x < 1$. The answer is $\boxed{\textbf{(E)}}$.

See Also

1963 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
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