logarithmic function help!!!!

Question

The graph below shows the logarithmic function . identify the domain and Range.

- What is the value of the x-intercept?

-Explain why the x-intercept of a logarithmic function will always have this value

-In details why a logarithmic function will never have a y- intercept.

-explain why (8,3) exists as a point on this graph based on your knowledge of exponential function.

Answer 1

(a)

The function is log₂(x).

Let y = log₂(x)

As the logarithmic function is not defined for the negative values of x.

Then domain of the function is (0, ∞)

Range of the function is the value of function for different values of x.

Range of the function is (-∞, ∞).

Therefore,

Domain is (0, ∞).

Range is (-∞, ∞).

Answer 2

(b)

The function is log₂(x).

Let y = log₂(x)

To find the x - intercept of the function, substitute y = 0 in the function.

log₂(x) = 0

log(x)/log(2) = 0                (Using logaritmic properties log_a(b) = log(b)/log(a))

log(x) = 0

log(x) = log(1)

Cancel the logarithm on each sides.

x = 1

Therefore the x - intercept of the function is x = 1.

That is why x intercept of logarithmic function always has a value x = 1.

Answer 3

(c)

The function is log₂(x).

Let y = log₂(x)

To find the y intercept of the function, we need to substitute x = 0 in the function.

we know that the value of loagrithmic function at x = 0 is undefined.

Therefore there is no y intercept.

Answer 4

(d)

The function is log₂(x).

let y = log₂(x)

Now let us consider x = 8

x = 8 is in the domain of function

y = log₂(8)

y = log₂(2³)

y =3*log₂(2)

y = 3.             (log₂(2) = 1)

y = 3 is in the range of function

Point (8,3) exists.

Graph

Therefore we can observe that the point (8,3) exists.