If x and log_10x are real numbers and log_10x<0, then:

1963 AMC 12 Problem 5

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x

and

\log_{10}x

are real numbers and

\log_{10}x\lt0,

then:

x\lt0

-1\lt x\lt1

0\lt x\leq1

-1\lt x\lt0

0\lt x\lt1

A real logarithm requires a positive argument

Compare

x

with

10^0

The logarithm is real only for

x\gt0.

Since base

10

is greater than

1,

\log_{10}x\lt0=\log_{10}1

implies

x\lt1.

Thus, the correct answer is E.