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a,b,x satisfy logarithmic equation

Source: 1976 AHSME Problem 20

May 18, 2014
logarithmsquadraticsAMC

Problem Statement

Let a, b,a,~b, and xx be positive real numbers distinct from one. Then 4(log⁡ax)2+3(log⁡bx)2=8(log⁡ax)(log⁡bx)4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)
<spanclass=′latex−bold′>(A)</span>for all values of a, b, and x<span class='latex-bold'>(A) </span>\text{for all values of }a,~b,\text{ and }x\qquad
<spanclass=′latex−bold′>(B)</span>if and only if a=b2<span class='latex-bold'>(B) </span>\text{if and only if }a=b^2\qquad
<spanclass=′latex−bold′>(C)</span>if and only if b=a2<span class='latex-bold'>(C) </span>\text{if and only if }b=a^2\qquad
<spanclass=′latex−bold′>(D)</span>if and only if x=ab<span class='latex-bold'>(D) </span>\text{if and only if }x=ab\qquad
<spanclass=′latex−bold′>(E)</span>for none of these <span class='latex-bold'>(E) </span>\text{for none of these}
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