# How do you graph y=log_5(x+1)+1?

Sep 30, 2016

Check the explanation.

#### Explanation:

Assuiming we know the graph of $y = {\log}_{5} \left(x\right)$, we can see that your function comes from the forementioned one, with two changes:

${\log}_{5} \left(x\right) \setminus \to {\log}_{5} \left(x + 1\right) \setminus \to {\log}_{5} \left(x + 1\right) + 1$

The first change is a change of the type

$f \left(x\right) \setminus \to f \left(x + k\right)$

so we are adding a constant inside the argument of the function. This changes cause a horizontal shift, to the left if $k$ is positive, and to the right if $k$ is negative.

The second change is like

$f \left(x\right) \setminus \to f \left(x\right) + 1$

so the argument is unchanged, and we add a constant to the whole expression. These kind of manipulations cause a vertical shift, upwards if $k$ is positive, downwards if it's negative.

So, in your case, start from $y = {\log}_{5} \left(x\right)$ (which, I repeat, I'm taking for granted):
graph{log(x)/log(5) [-3.55, 16.45, -3.92, 6.08]}
Shift everything to the left by $1$ unit to obtain $y = {\log}_{5} \left(x + 1\right)$:
graph{log(x+1)/log(5) [-3.55, 16.45, -3.92, 6.08]}
Shift everything up by $1$ unit to obtain $y = {\log}_{5} \left(x + 1\right) + 1$:
graph{log(x+1)/log(5)+1 [-3.55, 16.45, -3.92, 6.08]}