1976 AMC 12 Problem 20

Question

Let

a,

b

and

x

be positive real numbers distinct from one. Then

\begin{aligned} 4(\log_a x)^2&+3(\log_b x)^2\\ &=8(\log_a x)(\log_b x) \end{aligned}

for all values of

a,

b

and

x

if and only if

a=b^2

if and only if

b=a^2

if and only if

x=ab

none of these

Move all terms to one side and factor the quadratic in the two logarithms

Translate each factor equation into a relation between the bases using change of base

Factoring gives

\begin{aligned} &\bigl(2\log_a x-\log_b x\bigr)\\ &\quad\cdot \bigl(2\log_a x-3\log_b x\bigr)=0. \end{aligned}

Since

x\ne1,

change of base shows that the first factor vanishes when

a=b^2,

while the second vanishes when

a^3=b^2.

Thus the equation holds under either of two conditions, and no listed “if and only if” statement describes their union.

Therefore, the correct answer is E.

Problem 20 in other years