11 ALGEBRA

2. Inverse of a Logarithmic Function

Recall that we defined logarithms so that we could write the inverse of ax in a convenient way. Thus the logarithmic function f(x) = loga x and the exponential function g(x) = ax are inverses of each other. This is why their graphs are reflections of each other in the line y = x, as we can see below.

We can also prove this inverse property formally, as follows:

By the property of an inverse function, the functions f: (0, ) , f(x) = loga x and g: (0, ), g(x) = ax are inverse functions if their composition is the identity function I(x). In other words,

(f g)(x) = (g f )(x) = I(x) = x f(x) = g–1(x) and g(x) = f –1(x).

Let us check that this is true.

(f g)(x) = f(g(x)) = (loga x) (ax) = loga(ax) = x loga a = x for all x , and (g f )(x) = g(f(x)) = (ax) (loga x) = a(loga x) = x for all x (0, ).

So f(x) = loga x and g(x) = ax are indeed inverse functions.

Note that as a result, f(x) = log x and g(x) = 10x are inverse functions. f(x) = ln x and g(x) = ex are also inverse functions.

Since only bijective functions have an inverse, and logarithmic and exponential functions are inverses of each other, we can conclude that logarithmic functions are bijective. In other words, logarithmic functions are both one-to-one and onto.

We can use the one-to-one property of logarithmic functions to solve equations involving logarithms. For now it is enough to state that

loga x = loga y x = y .

For example, if log2 x = log2 5 then x = 5. We will look at this property in more detail in Chapter 3.

EXAMPLE 57 Find the inverse of f: (0, ), f(x) = 3x+1.

Solution y = 3x+1 x + 1 = log3 y x = log3 y – 1.

Interchanging x and y gives us the inverse of the given function: f –1: (0, ) , f –1(x) = log3 x – 1.

EXAMPLE 59 Find the inverse of each function.

a. f(x) = 1 – 31 – 2x b. f(x) = ln(x + 2)

Check Yourself 11

3. Monotone Property of Logarithmic Functions

The graphs of logarithmic functions suggest the following properties:

If a 1 then f(x) = loga x is strictly increasing.

If 0 a 1 then f(x) = loga x is strictly decreasing.

We will look at logarithmic inequalities in more detail in the next chapter. For now, it is enough to remember two important rules:

1. Since f(x) = loga x is an increasing function when a > 1, we can write

x1 x2 f(x1) f(x2), i.e. x1 < x2 loga x1 loga x2 for a 1.

In other words, if we take the logarithm of both sides of an inequality to the same base a > 1, the direction of the inequality will stay the same.

For example, 2 < 3, so log5 2 < log5 3 because a = 5 > 1.

2. Since f(x) = loga x is strictly decreasing when 0 < a < 1, we have

x1 x2 f(x1) f(x2), i.e. x1 x2 loga x1 loga x1.

In other words, if we take the logarithms of both sides of an inequality to the same base a (0, 1), the direction of the inequality will be reversed.

Check Yourself 12

4. Determine whether each number is positive or negative.

LOGARITHMIC SPIRALS

A logarithmic spiral is a special kind of spiral curve which often appears in nature. It is defined by the polar equation r = a eb where r is the distance of the curve from the origin, is the angle of the curve to the x-axis, and a and b are arbitrary constants. The logarithmic relation between the angle of the spiral and its radius ( = 1b ln ar ) gives the spiral its name. The logarithmic spiral is also known as the equiangular spiral or growth spiral.

C.SIMPLE VARIATIONS OF LOGARITHMIC FUNCTIONS (OPTIONAL)

Now that we are familiar with basic logarithmic functions and their graphs, we are ready to study the properties and graphs of general logarithmic functions.

A function of the form f(x) = c loga[d(x + p)] + k where a, c, p, d and k are real numbers with c, d 0 is called a logarithmic function with base a.

We can sketch the graph of a logarithmic function by plotting a selection of points, but this requires complicated calculations for finding the points. Alternatively, we can use some simple strategies to sketch the graph of a general logarithmic function with little or no computation. Let us first recall the common properties of all basic logarithmic functions (f(x) = logax):

1.The domain is (0, ).

2.The range is .

3.The graph does not cross the y-axis. The y-axis is a vertical asymptote.

4.The graph has an x-intercept at (1, 0).

5.(a, 1) is also a point on the graph.

6.The function is either strictly increasing (for a > 1) or strictly decreasing (for a (0, 1)):

We can use all of these observations to roughly sketch the graph of any basic logarithmic function f(x) = loga x.

One method for drawing a rough graph of a general logarithmic function g(x) = c loga[d(x + p)] + k is to start with the graph of f(x) = logax and apply transformations (horizontal or vertical shift, shrink, stretch or reflection) to f(x) step by step. This is similar to the approach that we used to sketch the graph of exponential functions.

We can summarize the main transformations as follows:

For a function g(x) = c loga[d(x + p)] + k, the constants c, a, d, p and k have the following effect on the graph of f(x) = loga x:

k represents a vertical shift k units up if k > 0, or |k| units down if k < 0.

p represents a horizontal shift p units to the left if p > 0, or |p| units to the right if p < 0.

c represents a vertical stretch, shrink or reflection:

–c < 0 means a reflection in the x-axis.

–|c| > 1 means a vertical stretch by a factor of |c|.

–0 < |c| < 1 means a vertical shrink by a factor of |c|.

d represents a horizontal stretch, shrink or reflection:

–d < 0 means a reflection in the y-axis.

–0 < |d| < 1 means a horizontal stretch by a factor of |d1|.

–|d| > 1 means a horizontal stretch by a factor of |d1|.

We mostly use these transformations to sketch the graphs of logarithmic functions containing absolute values.

We can alternatively sketch the graph of g(x) = c loga[d(x + p)] + k by identifying the vertical asymptote x = –p and two points which lie on the graph. To identify the points, we

The following table is helpful when using this second approach to sketching a graph:

Graphs of functions of the form f(x) = c loga[d(x + p)] + k, where d 0