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Difference between revisions of "1976 AHSME Problems/Problem 20"

(Created page with "Let <math>a,~b</math>, and <math>x</math> be positive real numbers distinct from one. Then <math>4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)</math> <math>\textbf{(A) }\text...")
 
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Because <math>\log_mn = \dfrac{\log{n}}{\log{m}}</math>, <math>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}</math>.
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Because <math>\log_{m} n = \dfrac{\log n}{\log m}</math>, <math>4(\log_{a} x)^2+3(\log_{b} x)^2 = </math> <math> \dfrac{4(\log x)^2}{(\log a)^2}+\dfrac{3(\log x)^2}{(\log b)^2} = \dfrac{(\log x)^2(4(\log a)^2+3(\log b)^2)}{(\log a \log b)^2}</math>.
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Therefore,
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<cmath>\dfrac{(\log x)^2(4(\log a)^2+3(\log b)^2)}{(\log a \log b)^2} = \frac{8(\log x)^2}{\log a \log b}</cmath>
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<cmath>\dfrac{4(\log a)^2+3(\log b)^2}{\log a \log b} = 8</cmath>
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<cmath>\dfrac{4(\log a)^2}{\log a \log b} + \dfrac{3(\log b)^2}{\log a \log b} = 8</cmath>
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<cmath>4 \log_{b} a + 3 \log_{a} b = 8</cmath>
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Now let <math>k = \log_{b} a</math>. Our equation becomes
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<cmath>4k + \frac{3}{k} = 8</cmath>
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<cmath>4k^2 - 8k + 3 = 0</cmath>
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<cmath>k = \frac{8 \pm \sqrt{64-4(4)(3)}}{8}</cmath>
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<cmath>k = \frac{1}{2}, \frac{3}{2}</cmath>
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Therefore, either <math>b = a^2</math> or <math>b = \sqrt[3] {a^2}</math>. Since no option matches both of these solutions, the answer is <math>\boxed{\textbf{(E)}}</math>.
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- mako17

Latest revision as of 03:43, 27 November 2021

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 $\log_{m} n = \dfrac{\log n}{\log m}$, $4(\log_{a} x)^2+3(\log_{b} x)^2 =$ $\dfrac{4(\log x)^2}{(\log a)^2}+\dfrac{3(\log x)^2}{(\log b)^2} = \dfrac{(\log x)^2(4(\log a)^2+3(\log b)^2)}{(\log a \log b)^2}$.

Therefore,

\[\dfrac{(\log x)^2(4(\log a)^2+3(\log b)^2)}{(\log a \log b)^2} = \frac{8(\log x)^2}{\log a \log b}\] \[\dfrac{4(\log a)^2+3(\log b)^2}{\log a \log b} = 8\]

\[\dfrac{4(\log a)^2}{\log a \log b} + \dfrac{3(\log b)^2}{\log a \log b} = 8\]

\[4 \log_{b} a + 3 \log_{a} b = 8\]

Now let $k = \log_{b} a$. Our equation becomes

\[4k + \frac{3}{k} = 8\] \[4k^2 - 8k + 3 = 0\] \[k = \frac{8 \pm \sqrt{64-4(4)(3)}}{8}\] \[k = \frac{1}{2}, \frac{3}{2}\]

Therefore, either $b = a^2$ or $b = \sqrt[3] {a^2}$. Since no option matches both of these solutions, the answer is $\boxed{\textbf{(E)}}$.

- mako17

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