If a>1, b>1 and p=(log_b(log_ba))/(log_ba), then a^p equals

1982 AMC 12 Problem 13

Attempt Problem 13 of the 1982 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 1982 AMC 12 solutions, or check the answer key.

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13.

a\gt1,

b\gt1

and

p=\frac{\log_b(\log_ba)}{\log_ba},

then

a^p

equals

1

b

\log_ab

\log_ba

a^{\log_ba}

Let

t=\log_ba,

a=b^t

Rewrite

p

\frac{\log_bt}{t}

Set

t=\log_ba,

a=b^t

and

p=\frac{\log_bt}{t}.

Then

a^p=(b^t)^{\frac{\log_bt}{t}},

which equals

b^{\log_bt}=t.

Therefore

a^p=\log_ba.

Therefore, the correct answer is D.