Chapter 4 - Section 4.5 - Exponential and Logarithmic Equations - 4.5 Exercises - Page 446: 32

$x=\ln\frac{1}{3}$

Let $u=e^x$ $3e^{2x}+2e^x-1=0$ $3u^2+2u-1=0$ Factor. $(3u-1)(u+1)=0$ Equate each factor to 0. $u=\frac{1}{3}$ or $u=-1$ Since $e^x\gt0$ for all values of $x$ Choose only: $u=\frac{1}{3}=e^x$ Take the natural log of both sides. $x=\ln\frac{1}{3}$