# How do you find the domain and range of ln(14t)?

Dec 16, 2017

Domain: $x \in {\mathbb{R}}^{+}$
Range: $y \in \mathbb{R}$

#### Explanation:

The $\ln$ function is not defined for values less than or equal to $0$, so we can say that the domain is the positive real numbers, ${\mathbb{R}}^{+}$.

The range is a bit tricky. If you were looking at a graph of the function, you might think that you see a horizontal asymptote, but it actually isn't. It is just the fact that the $\ln \left(\right)$ function is increasing so slow that it seems to be approach some value, but if you look at the values as they go towards infinity, there is no bound.

You can use a bit of calculus to make this extra clear. If you look a the following limit,

${\lim}_{x \to \infty} \ln \left(14 x\right) = \infty$

it becomes quite clear that there is no bound and therefor no asymptote. This means that the range is all real numbers, $\mathbb{R}$.