Graphs of Logarithmic Functions and their Features

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.

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.

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.

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

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 1 2 3 4 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.

For the graph of y = , 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. 4 2 2 y x 1 2 3 4 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.

y 2 2 4 x 1 2 3 4 4 2 2 y x 1 2 3 4 y = log2 x In fact, the above graphs are typical graphs of y = loga x for a > 1 and 0 < a < 1.

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 1 2 3 4 x 1 2 3 4 In fact, the above graphs are typical graphs of y = loga x for a > 1 and 0 < a < 1.

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.

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.

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

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)

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

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.

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

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.

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

Relationship between Graphs of Exponential and Logarithmic Functions y 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

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