Question

What is the true solution to the equation below?

l n e Superscript l n x Baseline + l n e Superscript l n x squared Baseline = 2 l n 8
x = 2
x = 4
x = 8
x = 64

Answer 1

Interpret the equation as ln(e^{ln x}) + ln(e^{ln x^2}) = 2 ln 8, with x>0.

ln(e^{ln x}) = ln x and ln(e^{ln x^2}) = ln(x^2) = 2 ln x, so
3 ln x = 2 ln 8.

Thus ln x = (2/3) ln 8 = (2/3)(3 ln 2) = 2 ln 2, so x = e^{2 ln 2} = 2^2 = 4.

Answer: x = 4.