2022 AMC 12B Problem 16

Attempt Problem 16 of the 2022 AMC 12B 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 2022 AMC 12B solutions, or check the answer key.

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16.

Suppose

x

and

y

are positive real numbers such that

x^y = 2^{64}

and

(\log_2 x)^{\log_2 y} = 2^7.

What is the greatest possible value of

\log_2 y?

3

4

3 + \sqrt2

4 + \sqrt3

7

Let

a = \log_2 x

and

b = \log_2 y;

take

\log_2

of both equations

You get

a \cdot 2^b = 2^6

and

a = 2^{\frac{7}{b}};

combine to form a quadratic in

b

Let

a = \log_2 x

and

b = \log_2 y.

Taking

\log_2

x^y = 2^{64}

gives

y \log_2 x = 64,

i.e.

a \cdot 2^b = 2^6.

Taking

\log_2

of the second equation gives

b \log_2 a = 7,

a = 2^{\frac{7}{b}}.

Substituting,

2^{\frac{7}{b}} \cdot 2^b = 2^6,

b + \dfrac7b = 6,

i.e.

b^2 - 6b + 7 = 0.

Thus

b = 3 \pm \sqrt2,

and the greatest value of

\log_2 y

3 + \sqrt2.

Thus, the correct answer is C.

Problem 16 in other years