# What is the domain and range of y = log2^x?

Dec 13, 2017

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

#### Explanation:

To answer this question we must consider our log laws:

$\alpha \log \beta = \log {\beta}^{\alpha}$

So using the knowledge:

$y = \log {2}^{x} \implies y = x \log 2$

Now this is just linear!

We know $\log 2 \approx 0.301$

$\implies y = 0.301 x$

Now we see by a sketch:

graph{y = 0.301x [-10, 10, -5, 5]}

That all $x$ and all $y$ are defined, yielding:

$x \in \mathbb{R}$ and $y \in \mathbb{R}$