For how many integers n does the expression √(log(n^2)-(log n)^2)/(log n-3) represent a real number, where log denotes the base 10 logarithm?

2023 AMC 12B Problem 7

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For how many integers

n

does the expression

\sqrt{\frac{\log(n^2)-(\log n)^2}{\log n-3}}

represent a real number, where

\log

denotes the base

10

logarithm?

900

3

902

2

901

The expression is real exactly when the fraction under the square root is

\ge 0

Let

L=\log n;

the fraction is

\dfrac{L(2-L)}{L-3}\ge 0,

and

n

must be a positive integer

Write

L=\log n.

Then

\log(n^2)-(\log n)^2

=2L-L^2

=L(2-L),

and the fraction is

\dfrac{L(2-L)}{L-3}.

A sign chart shows this is

\ge 0

exactly when

L\le 0

2\le L\lt 3.

Since

n

is a positive integer,

L\le 0

forces

n=1,

while

2\le L\lt 3

gives

100\le n\le 999,

which is

900

values. In total

1+900=901.

Thus, the correct answer is E.