AMC 10 · 2020 · #13

Grade 11 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
logarithm-propertiesnested-radical-denestingexponentsperfect-squares convert-to-algebraidentify-subproblems ↑ Prerequisites: logarithm-propertiesexponents
📏 Medium solution 💡 3 insights
Problem
Find the exact value of the square root of the sum of the base-two logarithm of six and the base-three logarithm of six, and say which of the five listed expressions it equals. The two logarithms have different bases but the same argument.

Pick an answer.

(A)
1
(B)
$\sqrt{\log_5{6}}$
(C)
2
(D)
$\sqrt{\log_2{3}}+\sqrt{\log_3{2}}$
(E)
$\sqrt{\log_2{6}}+\sqrt{\log_3{6}}$
How to solve
Strategy Introduce a Variable

Nothing simplifies while the two logarithms stay locked in different bases, so the first move is to split each one with the product rule; that peels off a plain 1 from each and leaves only log₂ 3 and log₃ 2. Those two leftovers are reciprocals, which is easy to see once they carry short names a and b, and the fact ab=1 is exactly what turns the constant 2 into 2√(ab). With that substitution the quantity under the root reads a+2√(ab)+b, a perfect square in disguise, so the square root comes off cleanly instead of being estimated. Naming the pieces is what makes the hidden square visible.

1STEP 1

Split with the product rule

Each becomes one plus something.

log₂ 6+log₃ 6=(1+log₂ 3)+(1+log₃ 2)=2+log₂ 3+log₃ 2
2STEP 2

Name the two leftovers

They are reciprocals, so their product is one.

a=log₂ 3, b=log₃ 2=1/a, ab=1, log₂ 6+log₃ 6=a+b+2
3STEP 3

Spot the perfect square

The sum becomes a perfect square.

a+b+2=a+2√(ab)+b=(√(a)+√(b))²
4STEP 4

Take the root

Taking the root leaves a sum of two square roots.

√(log₂ 6+log₃ 6)=√((√(a)+√(b))²)=√(a)+√(b)=√(log₂ 3)+√(log₃ 2)
Answer
√(log₂{3})+√(log₃{2})
A quick numerical check confirms the match. log₂ 3≈ 1.585 and log₃ 2≈ 0.631, so the quantity under the root is about 2+1.585+0.631=4.216 and its square root is about 2.053. Choice (D) gives √(1.585)+√(0.631)≈ 1.259+0.794≈ 2.053, the same value. The other choices miss: 1 and 2 are too small, √(log₅ 6)≈ 1.055 is far too small, and √(log₂ 6)+√(log₃ 6)≈ 1.608+1.277≈ 2.885 is too large. That last one is the trap of splitting a root across a sum; √(x+y) is never √(x)+√(y) for positive x and y, since squaring the right side leaves an extra 2√(xy).
💡Key takeaway

When two numbers multiply to 1, their sum plus 2 is a perfect square, so look for (√(a)+√(b))² before reaching for a calculator.

  • Split each log with the product rule
  • Name the two leftovers
  • Spot the hidden perfect square
  • Take the root and match a choice

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