1984 AMC 12 Problem 14

Attempt Problem 14 of the 1984 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1984 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

14.

The product of all real roots of the equation

x^{\log_{10}x}=10

1

-1

10

10^{-1}

none of these

Take the base-

10

logarithm of both sides

Let

t=\log_{10}x

and solve the resulting equation in

t

The logarithm requires

x\gt0.

Taking base-

10

logarithms gives

(\log_{10}x)^2=1.

Thus

\log_{10}x=\pm1,

so the two roots are

10

and

10^{-1},

whose product is

1.

Therefore, the correct answer is A.

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