# What is the domain and range of y=ln(x^2)?

Apr 9, 2017

Domain for $y = \ln \left({x}^{2}\right)$ is $x \in R$ but $x \ne 0$, in other words $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$ and range is $\left(- \infty , \infty\right)$.
We cannot have logarithm of a number less than or equal to zero. As ${x}^{2}$ is always positive, only value not permissible is $0$.
Hence domain for $y = \ln \left({x}^{2}\right)$ is $x \in R$ but $x \ne 0$, in other words $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$
but as $x \to 0$, $\ln \left({x}^{2}\right) \to - \infty$, $y$ can take any value from $- \infty$ ao $\infty$ i.e. range is $\left(- \infty , \infty\right)$.