Question

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

Answer 1

Check the explanation.

Explanation:

Assuiming we know the graph of `y=log_5(x)`, we can see that your function comes from the forementioned one, with two changes:

`log_5(x) \to log_5(x+1) \to log_5(x+1)+1`

The first change is a change of the type

`f(x) \to f(x+k)`

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(x) \to f(x)+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(x)` (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(x+1)`:
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(x+1)+1`:
graph{log(x+1)/log(5)+1 [-3.55, 16.45, -3.92, 6.08]}