# How do you graph y=log_2 (x-2)?

Dec 10, 2017

#### Explanation:

We know how to sketch ${\log}_{2} x$ rather simply:

graph{lnx [-4.04, 15.96, -4.88, 5.12]}

But ${\log}_{2} \left(x - 2\right)$ is just ${\log}_{2} x$ shifted $2$ to the right:

${\log}_{2} \left(x - 2\right)$: graph{ln(x-2) [-4.04, 15.96, -4.88, 5.12]}

The asymptote for ${\log}_{2} x$ is $x = 0$
The $x$ intercept is $1$

So the asymptote for ${\log}_{2} \left(x - 2\right)$ is $x = 2$
The new $x$ intercept is hence $3$

As indicated on the second graph

Dec 10, 2017

See below

#### Explanation:

$y = {\log}_{2} \left(x - 2\right)$

First, Let's find the domain:
$x - 2 > 0 \quad \implies \quad x > 2$

Now we know that this logaritmic function will be approching x=2 but will never get there.The base equals 2 which is greater than 1 so that means It's increasing its value on the whole domain.

Intercept x axis: [3,0]
We can find it by equation but it's much easier to simply think like this: $\mathmr{if} \quad \log x \quad \text{must equal 0 then x must equal 1} \quad$which means $x - 2 = 1 \quad \implies \quad x = 3$

Doesn't intercept y axis

Next point could be when y=1. (Let's do an equation this time)
$y = {\log}_{a} x \quad \implies \quad 1 = {\log}_{2} \left(x - 2\right) \quad$
${a}^{y} = x \quad \implies \quad {2}^{1} = x - 2$

$y = 1 \quad \implies \quad x = 4$