Question

Ln(x+2)-ln(4x+3)=ln(1/2*x)

Answer 1

ln(x+2)-ln(4x+3)=ln(1/2*x)

Using properties of logarithms


`(ln(x+2))/(ln(4x+3)) = ln (x)/(2) \\ \\ (x+2)/(4x+3) = (x)/(2) \\ \\2(x+2)=x(4x+3) 2x+4=4x^2+3x \\ \\ 4 x^(2) +x-4=0 \\ \\ x= \frac{-b+/- \sqrt{b^(2)-4ac} }{2a} \\ \\ x= ( -1+/-√(1+64) )/(8) \\ \\ x_(1) = (-1+ √(65) )/(8) \\ \\ x_(2) = (-1- √(65) )/(8) Check: When you substitute x_(2) into \\ \\ ln(4x+3)=ln(4* (-1- √(65) )/(8) ) =ln( (-1- √(65) )/(2) ) you will get negative number under ln, that is impossible ,`

so x2 is not a solution of this logarithmic equation.

Only x1 is a solution.