# Solve |ln(x + 3)| = 1. Give your answers in exact form.

Question
Equations
Solve $$\displaystyle{\left|{\ln{{\left({x}+{3}\right)}}}\right|}={1}$$. Give your answers in exact form.

2021-02-05
We are given $$\displaystyle{\mid}{\ln{{\left({x}+{3}{\mid}={1}\right.}}}$$
Since the left side is an absolute value expression and the right side is positive, there are two possible equations:
$$\displaystyle{\ln{{\left({x}+{3}\right)}}}=-{1}\ {\ln{{\left({x}+{3}\right)}}}={1}$$
$$\displaystyle{x}+{3}={e}^{{-{{1}}}}\ {x}+{3}={c}^{{1}}$$
$$\displaystyle{x}+{3}=\frac{{1}}{{c}}\ {x}+{3}={e}$$
$$\displaystyle{x}=\frac{{1}}{{c}}-{3}\ {x}={e}-{3}$$
$$\displaystyle{x}=\frac{{{1}-{3}{c}}}{{e}}$$
So, the solutions are:
$$\displaystyle{x}=\frac{{{1}-{3}{e}}}{{e}},{x}={e}-{3}$$

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