Answer
$x\approx 0.69 \text{ or } x =0$
Work Step by Step
Let
$u=e^x$
This means that:
$e^{2x} = u^2$
Thus, the given equation can be written as:
$u^2-3u+2=0$
Factor the trinomial. Look for factors of $2$ whose sum is $-3$.
These factors are $-2$ and $-1$. Therefore the factored form of the trinomial is:
$(u-2)(u-1)=0$
Use the zero product property by equating each factor to zero.
$u-2 = 0 \text{ or } u-1=0$
Solve each equation to obtain:
$u=2 \text{ or } u=1$
Switch $u$ back to $e^x$ to obtain:
$e^x=2 \text{ or } e^x=1$
Solve each equation by taking the natural logarithm of both sides.
$\ln{e^x} = 2 \text{ or } \ln{e^x}=1$
Use the rule $\ln{e^x} = x$ to obtain:
$x = \ln{2} \text{ or } x=\ln{1}$
Use a calculator to solve for $x$. Round-off answers to two decimal places.
$x\approx 0.69 \text{ or } x =0$
