2017 AMC 12A Problem 20

Attempt Problem 20 of the 2017 AMC 12A 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 2017 AMC 12A solutions, or check the answer key.

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20.

How many ordered pairs

(a,b)

such that

a

is a positive real number and

b

is an integer between

2

and

200,

inclusive, satisfy the equation

(\log_b a)^{2017}=\log_b(a^{2017})?

198

199

398

399

597

Let

u=\log_b a;

the equation becomes

u^{2017}=2017u

Either

u=0

u^{2016}=2017;

count the

a

values for each of the

199

allowed bases

Let

u=\log_b a.

Since

\log_b(a^{2017})=2017\log_b a,

the equation is

u^{2017}=2017u,

u=0

u^{2016}=2017.

u=0,

then

a=1,

valid for every one of the

199

bases. If

u^{2016}=2017,

then

u=\pm2017^{\frac{1}{2016}},

giving

2

values of

a

for each base, i.e.

2\cdot199=398

pairs.

In total there are

199+398=597

ordered pairs.

Thus, the correct answer is E.

Problem 20 in other years