What is the asymptote for the graph of this logarithmic function? f(x) = log3(x – 1)

Vertical Asymptote:

Horizontal asymptote:

it does not exist

Step-by-step explanation:

we are given

Vertical asymptote:

we know that vertical asymptotes are values of x where f(x) becomes +inf or -inf

we know that any log becomes -inf when value inside log is zero

so, we can set value inside log to zero

and then we can solve for x

we get

Horizontal asymptote:

we know that

horizontal asymptote is a value of y when x is +inf or -inf

For finding horizontal asymptote , we find lim x-->inf or -inf

so, it does not exist

The asymptote for the graph of this logarithmic function is at x = 1

An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. Whereas the logarithm is the inverse function to exponentiation and any exponential function can be expressed in logarithmic form.

First we solve

Raise both sides of the equation by the base of the logarithm:

Because any non-zero number raised to 0 equals 1, the right side simplifies to 1

By using the property of logarithms that where the left side simplifies to x-1

The equation is simply: x - 1 = 1

So x = 2.

The domain of a logarithmic function is the set of all positive real numbers. For example f(x) = log x also has an asymptote at x =0. But, since our function is log (x-1), we will move the asymptote to the right by 1 unit. Thus, x = 1.  Which explains that the graph will never touch at x=1 which will be the vertical asymptote.

Grade:  9

Subject:  mathematics

Chapter:  logarithmic function

Keywords:  logarithmic function, asymptote, graph, curve,  the inverse function