1991 AMC 12 Problem 24

Attempt Problem 24 of the 1991 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 1991 AMC 12 solutions, or check the answer key.

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24.

The graph,

G,

y=\log_{10}x

is rotated

90^\circ

counter-clockwise about the origin to obtain a new graph

G'.

Which of the following is an equation for

G'?

y=\log_{10}\left(\frac{x+90}{9}\right)

y=\log_x10

y=\frac1{x+1}

y=10^{-x}

y=10^x

90^\circ

counter-clockwise rotation sends

(x,y)

(-y,x)

Rename the rotated coordinates and solve the logarithmic relation for the new

y

A point

(u,\log_{10}u)

rotates to

(x,y)=(-\log_{10}u,u).

Hence

x=-\log_{10}y,

\log_{10}y=-x

and

y=10^{-x}.

Thus the correct answer is D.

Problem 24 in other years