Question

Find all real solutions to the logarithmic equation.
ln(x) = 2
e^3x = -1
ln(x) + ln(2) = 3
e^2x - 3e^x + 2 = 0
ln(x^2 - 1) - ln(x-1) = ln(4)
e^2x * e^3x - 3 = 2

Answer 1

The following tips may be of help:
1) \(\ln(a) = b ~~~~\text{implies}~~~~a = e^b\)
2) e^(anything) can never be less than or equal to zero.
3) \(\ln(A) + \ln(B) = \ln(AB)\). Then use tip #1
4) Make the substitution \(t = e^x\). The equation becomes \(t^2 - 3t + 2 = 0\). Solve for t. Replace t with e^x and solve for x.
5) \(\ln(A) - \ln(B) = \ln(A/B)\). Then use the fact that if ln(P) = ln(Q), then P = Q.
6) \(e^a * e^b = e^{a+b}\)