1976 AHSME Problems/Problem 20

Let $a,~b$, and $x$ be positive real numbers distinct from one. Then $4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)$

$\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad\\ \textbf{(B) }\text{if and only if }a=b^2\qquad\\ \textbf{(C) }\text{if and only if }b=a^2\qquad\\ \textbf{(D) }\text{if and only if }x=ab\qquad\\ \textbf{(E) }\text{for none of these}$


Solution

Because $\log_mn = \dfrac{\log{n}}{\log{m}}$, $4(\log_ax)^2+3(\log_bx)^2 = \dfrac{4(\logx)^2}{(\loga)^2}+\dfrac{3(\logx)^2}{(\logb)^2} = \dfrac{(\logx)^2(4(\loga)^2+3(\logb)^2)}{(\loga\logb)^2}$ (Error compiling LaTeX. ! Undefined control sequence.).

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