ln(3x+1)/ln(5x-1)=2

Question

solve the equation

Answer 1

ln(3x+1) / ln(5x-1)=2

e^ln(3x+1) / e^ln(5x-1)=e^2

(3x+1) /(5x-1)=e^2=7.3890560989306495

3x+1=7.389056(5x-1)=36.94528049465325x -7.389056

33.94528049465x=1+7.3890560989306495

x=8.3890560989/33.94528049465

x=0.24713468195408822

Answer 2

ln(3x+1)/ln(5x-1)=2

ln(3x+1)=2*ln(5x-1)

ln(3x+1)=ln{(5x-1)^2}

3x+1=(5x-1)^2

3x+1=(5x)^2-2*(5x)+1^2

3x+1=25x^2-10x+1

25x^2-13x=0

x(25x-13)=0

x=0 or x=13/25 (Discard x=0 because it does not satisfy the equation)

Therefore x = 13/25=0.52

CHECK TO SEE IF IT SATISFY THE EQUATION

ln(3*.52+1)/ln(5*.52) = 2

ln(2.56)/ln(1.6) = 2

2 = 2

It checks out