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

Mathematics

2 answers:

Pepsi [2]3 years ago

7 0

Answer:

Vertical Asymptote:

$x=1$

Horizontal asymptote:

it does not exist

Step-by-step explanation:

we are given

$f(x)=log_3(x-1)$

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

$x-1=0$

we get

$x=1$

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

$\lim_{x \to \infty} f(x)= \lim_{x \to \infty}log_3(x-1)$

$\lim_{x \to \infty} f(x)=log_3(\infty-1)$

$\lim_{x \to \infty} f(x)=undefined$

so, it does not exist

inessss [21]3 years ago

7 0

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

<h3>Further explanation </h3>

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 $log_3(x-1) = 0$

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

$3^(log_3(x-1)) = 3^0$

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

By using the property of logarithms that $b^{(logbx)} = x,$ 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.

<h3>Learn more</h3>