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  4. If 2( log 2 3)x=3( log 3 2)x then the value of x is equal to

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Q. If $2^{\left(\log _2 3\right)^x}=3^{\left(\log _3 2\right)^x}$ then the value of $x$ is equal to

Continuity and Differentiability

Solution:

$2^{\left(\log _2 3\right)^x}=3^{\left(\log _3 2\right)^x}$ Taking $\log$ to the base 2 on both the sides, we get $\left(\log _2 3\right)^x \cdot \log _2 2=\left(\log _3 2\right)^x \log _2 3$ $\left(\log _2 3\right)^{x-1}=\left(\log _3 2\right)^x \Rightarrow \frac{\left(\log _2 3\right)^{x-1}}{\left(\log _3 2\right)^x}=1$ $ \left(\log _2 3\right)^{2 x-1}=1=\left(\log _2 3\right)^0 $ $\Rightarrow 2 x-1=0 \Rightarrow x=\frac{1}{2}$