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  4. The number of solution of the equation 2-x+3 log 5 2= log 5(3x-52-x) is

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Q. The number of solution of the equation $2-x+3 \log _5 2=\log _5\left(3^x-5^{2-x}\right)$ is

Continuity and Differentiability

Solution:

$ 2- x +3 \log _5 2=\log _5\left(3^x-5^{2-x}\right)$
$\log _5 5^{2-x}+\log _5 2^3=\log _5\left(3^x-5^{2-x}\right) $
$5^{2-x} \cdot 8=3^x-5^{2-x}$
$5^{2-x}(8+1)=3^x$
$\frac{5^2}{5^x} \cdot 9=3^x $
$15^2=15^x \Rightarrow x=2$