Solve (e^x+5)/(9e^-x+4)=2

(e^x+5)/(9e^-x+4)=2  =  multiply both sides by (9e^-x+4)

e^x + 5   =  2 (9e^(-x) + 4)     simplify

e^x + 5  = 18e^(-x) + 8     

e^x - 3 - 18e^(-x)  = 0        multiply through by  e^x

e^2x - 3e^x - 18 = 0    factor

(e^x - 6) (e^x + 3)  = 0

Setting both facotrs to 0

e^x + 3  = 0      has no real solution

e^x - 6  =  0      add 6 to both sides

e^x =  6      take the natural log of both sides

ln e^x =  ln 6        and by a log property, we can write

x* lne = ln 6        and ln e  = 1 so we can ignore this

x =  ln 6