Logarithm (Equation)

(CIE 0606/2021/m/12/Q1)
Find the exact solutions of the equation $3(\ln 5x)^2+2\ln 5x-1=0.$ [4] 

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*********math solution*************

$\begin{array}{rcl}3(\ln 5x)^2+2\ln 5x-1&=&0\\ (3\ln 5x-1)(\ln 5x+1)&=&0\\3\ln 5x-1=0&\mbox{(or)}&\ln 5x+1=0\\\ln 5x=\dfrac 13&\mbox{(or)}&\ln 5x=-1\\5x=e^{\frac 13}&\mbox{(or)}&5x=e^{-1}\end{array}$

Hence $x=\dfrac 15e^{\frac 13}$ or $x=\dfrac{1}{5e}$.


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