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Logarithmic Functions
In this section we introduce logarithmic functions. Notice that every exponential function f ( x ) = ax , with a > 0 and a โ 1, is a one-to-one function by the Horizontal Line Test and therefore has an inverse function. The inverse function of the exponential function with base a is called the logarithmic function with base a and is denoted by log (^) a x. Recall that f -1^ is defined by
f โ^1 ( y ) = x โ f ( x )= y
This leads to the following definition of the logarithmic function.
Definition of the Logarithmic Function:
Let a be a positive number with a โ 0. The logarithmic function with base a , denoted by log (^) a , is defined by
log (^) a x = y โ a y = x
In other words, this says that
log (^) a x is the exponent to which the base a must be raised to give x.
The form log (^) a x = y is called the logarithmic form , and the form ay^ = x is called the exponential form. Notice that in both forms the base is the same:
Example 1: Express each equation in exponential form.
(a) log 49 7 = 2 (b) log 16 4 =^12
Solution:
From the definition of the logarithmic function we know
log (^) a x = y โ ay = x
This implies
(a) log 49 7 = 2 โ 7 2 = 49
(b) 1 12 log 16 4 = 2 โ 16 = 4
Example 2: Express each equation in logarithmic form.
(a) 34 = 81 (b) 6 โ^1 =^16
Solution:
From the definition of the logarithmic function we know
a y = x โ log ax = y
This implies
(a) 34 = 81 โ log 81 3 = 4
(b) 6 โ^1 = 16 โ log 616 = โ 1
Graphs of Logarithmic Functions:
Since the logarithmic function f ( x ) = log (^) a x is the inverse of the exponential function f ( x ) = ax , the graphs of these two functions are reflections of each other through the line y = x.
Example 3 (Continued):
Step 3: To find the graph of y = log (^) 5 x , all we need to do is reflect the graph of y = 5 x^ over the line y = x , because they are inverses.
Another way we can find the graph of y = log (^) 5 x is to take the chart we found in Step 1 for y = 5 x , and switch the x and y values. Then we plot the new points and draw a smooth curve connecting them.
The figure below shows the graphs of the family of logarithmic functions with bases 2, 3, 5, and 10.
We can now add the logarithmic function to our list of library functions. In addition, we can perform transformations to the logarithmic function using the techniques learned earlier.
Example 4: Graph the function f ( x ) = - log (^) 3 ( x + 2), not by plotting points, but by starting from the graphs in the above figure. State the domain, range, and asymptote.
Solution:
Step 1: To obtain the graph of f ( x ) = - log (^) 3 ( x + 2), we start with the graph of f ( x ) = log (^) 3 x , reflect it across the x -axis and shift it to the left 2 units.
Step 2: Notice that while the vertical asymptote is not actually part of the graph, it also shifts left 2 units, and so the vertical asymptote of f ( x ) = - log (^) 3 ( x + 2) is the line x = โ2. Looking at the graph, we see that the domain of f is (โ2, โ), and the range is รบ.
Some important properties of logarithms are as follows:
By the definition of inverse functions we have
ln x = y โ ey = x
The same important properties of logarithms that were listed above also apply to natural logarithms.
Properties of Natural Logarithms:
Example 5: Evaluate the expressions.
(a) log 1 7 (b) log 3 3 (c) ln e^12 (d) 10 log^ ฯ
Example 5 (Continued):
Solution (a):
The first property of logarithms says log (^) a 1 = 0. Thus,
log 1 7 = 0
Solution (b):
The second property of logarithms says log (^) a a = 1. Thus,
log 3 3 = 1.
Solution (c):
The third property of natural logarithms says ln ex^ = x. Thus,
ln e^12 = 12.
Solution (d):
Step 1: First note that log ฯ = log 10 ฯ. So
10 log ฯ^ = 10 log^10 ฯ
Step 2: The fourth property of logarithms says a log a^ x^ = x. Thus
10 log^10 ฯ^ = ฯ.
Example 6: Use the definition of the logarithmic function to find x.
(a) 3 =log 2 x (b) โ 4 =log 3 x (c) 4 =log 625 x (d) โ 2 =log 100 x
Solution (a):
Step 1: By the definition of the logarithm, we can rewrite the expression in exponential form.
3 3 = log 2 x โ 2 = x
Example 6 (Continued):
Step 2: Solve for x.
2
2 1 2 1 2 100 1 100 1 10
multiply both sides by^2
divide both sides by 100 take the square root of both sides
x
x
x
x x
x x
Again we note that a logarithm cannot have a negative base. So, we discard the extraneous solution x = โ 101 , and therefore x = 101 is the only solution to the expression โ 2 = log 100 x.
