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Question

If base of the log is 2, then equation X(3/4)(logX(5/4))=2 have:

A
at least one real solution
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B
exactly three real solutions
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C
exactly one irrational solution
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D
All of these
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Solution

The correct option is A at least one real solution
X(3/4)(logX5/4)=2
Take log base 2 on both sides.
log2X(3/4)(logX5/4)=log221/2
34(logX54).logX=12
log2X54.logX23=0
12log2X15logX8=0
logX=15±225+38424=15±60924
X=215+60924,21560924

Hence there are two irrational solutions of the equation.

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