Product of all the solution of equation x^{\left(log\right)_{10} x} = \left(\right. 100 + 2^{\sqrt{\left(log\right)_{2} 3}} - 3^{\sqrt{\left(log\right)

Engineering

Mathematics

Properties of Logarithm

Question

Product of all the solution of equation $x^{\log_{10} x} = (100 + 2^{\sqrt{\log_{2} 3}} - 3^{\sqrt{\log_{3} 2}}) x$ is

$\frac{1}{10}$

100

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Take log both sides       ( Here $a^{\sqrt{\log_{a} b}} = b^{\sqrt{\log_{b} a}}$ )
log₁₀x.log₁₀x = 2 + log₁₀x
⇒    t² – t – 2 = 0   Here t = log₁₀x
$∴$ t = 2, t = –1
$∴$ x = 10², x = 10^–1
$∴$    product of solution is 10² × $\frac{1}{10}$ = 10.