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1962 AHSME Problems/Problem 28

Problem

The set of $x$-values satisfying the equation $x^{\log_{10} x} = \frac{x^3}{100}$ consists of:

$\textbf{(A)}\ \frac{1}{10}\qquad\textbf{(B)}\ \text{10, only}\qquad\textbf{(C)}\ \text{100, only}\qquad\textbf{(D)}\ \text{10 or 100, only}\qquad\textbf{(E)}\ \text{more than two real numbers.}$


Solution

Taking the base-$x$ logarithm of both sides gives $\log_{10}x=\log_x\frac{x^3}{100}$. This simplifies to \[\log_{10}x=\log_x{x^3} - \log_x{100}\] \[\log_{10}x+\log_x{100}=3\] \[\log_{10}x+2 \log_x{10}=3\] At this point, we substitute $u=\log_{10}x$. \[u+\frac2u=3\] \[u^2+2=3u\] \[u^2-3u+2=0\] \[(u-2)(u-1)=0\] \[u\in\{1, 2\}\] \[x\in\{10, 100\}\] The answer is $\boxed{\textbf{(D)}}$.

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