FIND DOMAIN AND VERTICAL ASYMPTOTES OF LOGARITHMIC FUNCTION

For the following exercises, state the domain and the vertical asymptote of the function.

Problem 1 :

f(x) = log_b(x - 5)

Solution:

f(x) = log_b(x - 5)

Domain:

The domain is all values of x that make the expression defined.

f(x) = log_b(x - 5)

x - 5 > 0

x > 5

Domain: (5, ∞)

Vertical asymptote:

An asymptote is a line that a curve approaches but never touche.

f(x) = log_b(x - 5)

x - 5 = 0

x = 5

Vertical asymptote x = 5

Problem 2 :

g(x) = ln(3 - x)

Solution:

g(x) = ln(3 - x)

Domain:

3 - x > 0

3 > x

x < 3

Domain: (-∞, 3)

Vertical asymptote:

3 - x = 0

x = 3

vertical asymptote x = 3

Problem 3 :

f(x) = log(3x + 1)

Solution:

f(x) = log(3x + 1)

Domain:

3x + 1 > 0

x > -1/3

Domain: (-1/3, ∞)

Vertical asymptote:

3x + 1 = 0

3x = - 1

x = -1/3

vertical asymptote x = -1/3

Problem 4 :

f(x) = 3log(-x) + 2

Solution:

f(x) = 3log(-x) + 2

Domain:

-x > 0

x < 0

Domain: (-∞, 0)

Vertical asymptote:

-x = 0

x = 0

vertical asymptote x = 0

Problem 5 :

g(x) = -ln(3x + 9) - 7

Solution:

g(x) = -ln(3x + 9) - 7

Domain:

3x + 9 > 0

3x > -9

x > -3

Domain: (-3, ∞)

Vertical asymptote:

3x + 9 = 0

3x = -9

x = -3

vertical asymptote x = -3

Find the vertical asymptote, domain of each of the following logarithmic functions.

Problem 6 :

f(x) = log₂(x + 5) - 3

Solution:

f(x) = log₂(x + 5) - 3

Domain:

x + 5 > 0

x > -5

Domain: (-5, ∞)

Vertical asymptote:

x + 5 = 0

x = -5

vertical asymptote x = -5

Problem 7 :

f(x) = log₅(x - 3) + 1

Solution:

f(x) = log₅(x - 3) + 1

Domain:

x - 3 > 0

x > 3

Domain: (3, ∞)

Vertical asymptote:

x - 3 = 0

x = 3

vertical asymptote x = 3

Problem 8 :

f(x) = log₃(x - 4) + 2

Solution:

f(x) = log₃(x - 4) + 2

Domain:

x - 4 > 0

x > 4

Domain: (4, ∞)

Vertical asymptote:

x - 4 = 0

x = 4

vertical asymptote x = 4

Problem 9 :

f(x) = 3log₂(x - 1) + 2

Solution:

f(x) = 3log₂(x - 1) + 2

Domain:

x - 1 > 0

x > 1

Domain: (1, ∞)

Vertical asymptote:

x - 1 = 0

x = 1

vertical asymptote x = 1

Problem 10 :

f(x) = 1/2log₄(x - 6) - 5

Solution:

f(x) = 1/2log₄(x - 6) - 5

Domain:

x - 6 > 0

x > 6

Domain: (6, ∞)

Vertical asymptote:

x - 6 = 0

x = 6

vertical asymptote x = 6

Problem 11 :

f(x) = -4log₂(x - 2)

Solution:

f(x) = -4log₂(x - 2)

Domain:

x - 2 > 0

x > 2

Domain: (2, ∞)

Vertical asymptote:

x - 2 = 0

x = 2

vertical asymptote x = 2