1.Express (xe^x)^2=30e^(-x) in the form lnx=ax+b,and find the value of a and of b .

Question

2.solve 4(3^2x)=e^x

Answer 1

1. take ln of both sides

ln(x e^x)^2 = ln(30 e^-x)
2 lnx + 2ln e^x = ln30 + ln e^-x
2lnx + 2x = ln30 -x
2 lnx = -3x + ln30
lnx = (-3/2)x + (1/2)ln30 or (-3/2)x + ln (√30)
comparing this with ax + b
a = -3/2 , b = ln √30

2. take ln of both sides and use rules of logs

ln (4(3^2x) ) = ln e^x
ln4 + xln3 = x
x - 2xln3 = ln4
x(1 - 2ln3) = ln4
x = ln4/(1-2ln3) or appr. -1.1579

check:
LS = 4(3^-2.3158... = .31413784
RS = e^-1.157923.. = .31413784 , how about that?