Graphs of Logarithmic Functions
y = log a x represents the graph of a logarithmic function where x > 0, a > 0 and a ≠ 1.
Exponential Function
Logarithmic Function
f ( x ) = log a x
f ( x) = a x
f ( x) = 2 x
f ( x ) = log 2 x
The logarithmic and exponential functions are inverses. Therefore, they are symmetrical about the line y = x.
y=x
Properties of the Exponential Function
Domain: All Real Numbers
Range: y > 0
y‐intercept: (0, 1)
Horizontal Asymptote: y = 0
The function is increasing.
Properties of the Logarithmic Function
Domain: x > 0
Range: All Real Numbers
x‐intercept: (1, 0)
Vertical Asymptote: x = 0
The function is increasing.
f ( x) = 2 x
f ( x ) = log 2 x
Properties of y = log 2 x
Domain: x > 0
Range: All real numbers
y = log 2 x
x-intercept: (1, 0)
Vertical Asymptote: x = 0
y = log 4 x
The function is increasing.
Properties of y = log 4 x
Domain: x > 0
Range: All real numbers
x-intercept: (1, 0)
Vertical Asymptote: x = 0
The function is increasing.
The graph of y = log 4 x increases
more slowly and is closer to the
vertical asymptote.
Transformations of the Graph of the Logarithmic Function
y = log a ( − x )
Reflected over the vertical asymptote
Domain: x < 0
Range: All Real Numbers
x‐intercept: (‐1, 0)
Vertical Asymptote: x = 0
The function is decreasing.
f ( x) = log a ( − x )
f ( x ) = log a x
y = − log a x
Reflected over the known point
Domain: x > 0
Range: All Real Numbers
x‐intercept: (1, 0)
Vertical Asymptote: x = 0
The function is decreasing.
y = log a x
y = − log a x
y = log a ( x − h)
Shifted h units to the right
Domain: x > h
Range: All Real Numbers
x‐intercept: (1 + h, 0)
Vertical Asymptote: x = h
The function is increasing.
y = log a x
y = log a ( x − 2)
y = log a ( x + h)
Shifted h units to the left
Domain: x > h
Range: All Real Numbers
x‐intercept: (1 + h, 0)
Vertical Asymptote: x = h
The function is increasing.
y = log a ( x + 2)
y = log a x
y = log a ( x) + k
Shifted k units up
Domain: x > 0
Range: All Real Numbers
Vertical Asymptote: x = 0
The function is increasing.
y = log a ( x) + 2
y = log a x
y = log a ( x) − k
Shifted k units down
Domain: x > 0
Range: All Real Numbers
Vertical Asymptote: x = 0
The function is increasing.
y = log a x
y = log a ( x ) − 2
Directions: Graph each logarithmic function. Identify the domain, range, asymptotes, intercepts and determine if the function is increasing or decreasing.
1. y = log 4 ( x − 3)
Domain:
Range:
Asymptote:
y = log 4 x
y = log 4 ( x − 3)
1. y = log 4 ( x − 3)
Domain:
Range:
y = log 4 x
y = log 4 ( x − 3)
Asymptote:
Intercept:
Increasing/Decreasing:
2. y = − log 4 ( x − 2)
Domain:
y = log 4 x
Range:
Asymptote:
Intercept:
Increasing/Decreasing:
y = − log 4 ( x − 2)
3.
y = −3 − log 2 ( − x + 4)
y = log 2 x
Domain:
Range:
Asymptote:
Intercept:
Increasing/Decreasing:
y = −3 − log 2 ( − x + 4)