1. Graphs of Logarithmic Functions and their Features

2. Consider y = log x. x y 1 10 1
100 2 … …
For any value of x (x > 0), there is only one corresponding value of y.
y = log x is a function of x.

3. are logarithmic functions.
A function in the form y = loga x or f(x) = loga x, where a > 0 and a  1, is called a logarithmic function.
For example:
y = log3 x and
f(x) = x 2 1
log
are logarithmic functions.

4. The domain of a logarithmic function is all positive real numbers.
Since x = ay is positive (where a > 0), loga x is undefined for x  0.
The domain of a logarithmic function is all positive real numbers.

5. How do the graphs of logarithmic functions look like?
You can plot the graphs of y = log2 x and y = and see how
they look like. x 2 1
log

6. For the graph of y = log2 x, x 0.1 0.5 1 2 3 4 y 3.3 1 0 1 1.6 2
3.3 1 0 1 1.6 2
It lies on the right-hand side of the y-axis. y 2 2 4 x
y = log2 x
It has no maximum point, minimum point and axis of symmetry.
It cuts the
x-axis at
(1, 0).
As x increases, the value of y increases.

7. For the graph of y = , x 0.1 0.5 1 2 3 4 y 3.3 1 0 1 1.6 2
 1 1.6 2
It lies on the right-hand side of the y-axis. 4 2 2 y x
It has no maximum point, minimum point and axis of symmetry.
It cuts the
x-axis at
(1, 0).
As x increases, the value of y decreases.

8. y 2 2 4 x 4 2 2 y x
y = log2 x
In fact, the above graphs are typical graphs of y = loga x for a > 1 and 0 < a < 1.

9. y y
y = log a x
(a > 1)
y = log2 x 4 2 2
y = log a x
(0 < a < 1) 2 2 4 x x
In fact, the above graphs are typical graphs of y = loga x for a > 1 and 0 < a < 1.

10. Common features for the graph of y = loga x
(1, 0) x y
(1, 0) x y
1. The graphs cut the x-axis at (1, 0).
2. The graphs never cut the y-axis. They lie on the right-hand side of the y-axis.
3. The graphs have neither a maximum point, a minimum point nor an axis of symmetry.

11. Differences for the graph of y = loga x
(1, 0) x y
As x increases, the value of y increases.
As x increases, the value of y decreases.
(1, 0) x y
For 0 < x < 1,
y < 0.
For 0 < x < 1,
y > 0
As x increases, the rate of decrease of y becomes smaller.
As x increases, the rate of increase of y becomes smaller.
For x > 1,
y > 0.
For x > 1,
y < 0.

12. The table below summarizes the features of the graphs of logarithmic functions for a > 1 and 0 < a < 1.

13. Have you noticed any relation between the graphs of y = log2 x
and ?
The graph of
can be obtained by reflecting the graph of y = log2 x about the x-axis, and vice versa.
Q(2, 4)
P(1, 2)
P’(1, 2)
Q’(2, 4)

14. In general, the graphs of y = loga x and
show reflectional symmetry with each other about the x-axis.
axis of symmetry

15. Consider the following graphs.x y
y = log2 x
y = log5 x
y = log10 x
y = log0.5 x
y = log0.2 x
y = log0.1 x
The larger the value of a, the flatter is the graph of y = loga x.
0 < a < 1
The smaller the value of a, the flatter is the graph of y = loga x.

16. Follow-up question
In the figure, the graph of y = log4 x cuts the x-axis at A.
(a) Write down the coordinates of A.
(b) Sketch the graph of . A x y
y = log4 x

17. The graph of y = log4 x cuts the x-axis at (1, 0).A x y
The graph of y = log4 x cuts the x-axis at (1, 0).
y = log4 x
(a) The coordinates of A are (1, 0).
(b) The graph of can be obtained by
reflecting the graph of y = log4 x about the x-axis.

18. Have you noticed any relation between these two graphs?
Relationship between Graphs of Exponential and Logarithmic Functions x y
Have you noticed any relation between these two graphs?
y = 2x
Q’(1, 2)
P’(0, 1)
Q(2, 1)
P(1, 0)
y = log2 x

19. Relationship between Graphs of Exponential and Logarithmic Functionsy
y = 2x
The graphs of
y = 2x and y = log2 x show reflectional symmetry with each other about the line y = x.
Q’(1, 2)
P’(0, 1)
Q(2, 1)
P(1, 0)
y = x
y = log2 x

20. Similarly, the graphs of and
show reflectional symmetry with each other about the line y = x. x y
y = x