2024 AMC 12B Problems/Problem 8

Problem

What value of $x$ satisfies \[\frac{\log_2x \cdot \log_3x}{\log_2x+\log_3x}=2?\]

$\textbf{(A) } 25 \qquad\textbf{(B) } 32 \qquad\textbf{(C) } 36 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 48$

Solution 1

We have log2xlog3x=2(log2x+log3x)1=2(log2x+log3x)log2xlog3x1=2(1log3x+1log2x)1=2(logx3+logx2)logx6=12x12=6x=36 so $\boxed{\textbf{(C) }36}$

~kafuu_chino

Solution 2 (Change of Base)

log2xlog3xlog2x+log3x=2log2xlog3x=2(log2x+log3x)log2xlog3x=2log2x+2log3xlogxlog2logxlog3=2logxlog2+2logxlog3(logx)2log2log3=2logxlog3+2logxlog2log2log3(logx)2=2logxlog3+2logxlog2(logx)2=2logx(log2+log3)logx=2(log2+log3)x=102(log2+log3)x=(10log210log3)2x=(23)2=62=(C) 36

~sourodeepdeb

See also

2024 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png