# How do you graph y=ln abs(x+3)?

Nov 11, 2016

#### Explanation:

Note that $\ln \left(x + 3\right)$ is not defined if $x + 3 < 0$ i.e. $x < - 3$.

Further if $x = - 2$, $g \left(- 2\right) = \ln \left(- 2 + 3\right) = \ln 1 = 0$ and therefore in the interval $\left(- 2 , \infty\right)$ $\ln \left(x + 3\right)$ is above $x$-axis and in interval $\left(- 3 , - 2\right)$, it is below $x$-axis.

Also $g ' \left(x\right) = \frac{1}{x + 3}$ and hence in the interval $\left(- 3 , \infty\right)$, $g ' \left(x\right)$ is always positive and hence $g \left(x\right)$ is a continuously increasing function and its graph looks like
graph{ln(x+3) [-10, 10, -5, 5]}

Hence, for $y = | \ln \left(x + 3\right) |$,

While in the interval $\left(- 2 , \infty\right)$, it is exactly same as for $\ln \left(x + 3\right)$, in interval $\left(- 3 , - 2\right)$ it will be positive too and a reflection of curve above. Obviously, we have a discontinuity at $x = - 2$
graph{|ln(x+3)| [-10, 10, -5, 5]}