1. 4.3 Logarithmic Functions Logarithms Logarithmic Equations
Properties of Logarithms

2. Logarithms
The previous section dealt with exponential functions of the form y = ax where a ≠ 1 and a>0. The horizontal line test shows that exponential functions are one-to-one, and thus have inverse functions.
Starting with y = ax and interchanging x and y yields x = ay .Here we need to solve for y

3. Logarithms
Here y is the exponent. We call this exponent a logarithm, symbolized by “log.” We wrte y=loga x . The number a is called the base of the logarithm, and x is called the argument of the expression. It is read “logarithm with base a of x,” or “logarithm of x with base a.”

4. Logarithm
For all real numbers y and all positive numbers a and x, where a ≠ 1,
if and only if
A logarithm is an exponent. The expression
loga x represents the exponent to which the base “a” must be raised in order to obtain x.

5. Logarithms Logarithmic form: y = loga x Exponential form: ay = x
Base
Exponent
Exponential form: ay = x
Base

6. Logarithms

7. Solve a. Solution SOLVING LOGARITHMIC EQUATIONS Example 1
Write in exponential form.
Take cube roots

8. Solve b. Solution SOLVING LOGARITHMIC EQUATIONS Example 1
Write in exponential form.
The solution set is {32}.

9. Solve c. Solution SOLVING LOGARITHMIC EQUATIONS Example 1
Write in exponential form.
Write with the same base.
Power rule for exponents.
The solution set is
Set exponents equal.
Divide by 2.

10. Logarithmic Function
If a > 0, a ≠ 1, and x > 0, then
defines the logarithmic function with base a.

11. Logarithmic Function
Exponential and logarithmic functions are inverses of each other. The graph of
y = 2x is shown in red. The graph of its inverse is found by reflecting the graph across the line y = x.

12. Logarithmic Function
The graph of the inverse function, defined by y = log2 x, shown in blue, has the y-axis as a vertical asymptote.

13. Logarithmic Function
Functions
Domain
Range

14. LOGARITHMIC FUNCTION For (x) = log2 x: x (x) ¼ – 2 ½ – 1 1 2 4 8 32 4 8 3
(x) = loga x, a > 1, is increasing and continuous on its entire domain, (0, ) .
The y-axis is a vertical asymptote as x  0 from the right.

15. LOGARITHMIC FUNCTION For (x) = log1/2 x: x (x) ¼ 2 ½ 1 – 1 4 – 2 8
– 1 4
– 2 8
– 3
(x) = loga x, 0 < a < 1, is decreasing and continuous on its entire domain, (0, ) .
The y-axis is a vertical asymptote as x  0 from the right.

16. Graph the function. a. Solution GRAPHING LOGARITHMIC FUNCTIONS
Example 2
First graph y = (½)x . The graph of (x) = log1/2x is the reflection of the graph
y = (½)x across the line y = x.
Graph the function. a.
Solution

17. Graph the function. a. Solution GRAPHING LOGARITHMIC FUNCTIONS
Example 2
Graph the function. a.
Solution

18. Graph each function. Give the domain and range.
GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Example 3
Graph each function. Give the domain and range. a.
Solution
The graph of (x) = log2 (x – 1) is the graph of
(x) = log2 x translated 1 unit to the right. The vertical asymptote is x = 1. The domain of this function is (1, ) since logarithms can be found only for positive numbers. To find some ordered pairs to plot, use the equivalent exponential form of the equation
y = log2 (x – 1).

19. Solution GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS Example 3
We choose values for y and then calculate each of the corresponding x-values. The range is
(– , ).

20. Graph each function. Give the domain and range.
GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Example 3
Graph each function. Give the domain and range. b.
Solution
The function defined by (x) = (log3 x) – 1 has the same graph as g(x) = log3 x translated 1 unit down. We find ordered pairs to plot by writing y = (log3 x) – 1
in exponential form.

21. Graph each function. Give the domain and range.
GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Example 3
Graph each function. Give the domain and range. b.
Solution
Again, choose y-values and calculate the corresponding x-values. The domain is (0, ) and the range is (– , ).

22. Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real number r:
Property
Description
The logarithm of the product of two numbers is equal to the sum of the
logarithms of the numbers.
Product Property

23. Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real number r:
Property
Description
The logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers.
Quotient Property

24. Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real number r:
Property
Description
The logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number.
Power Property

25. Properties of Logarithms
Two additional properties of logarithms follow directly from the definition of loga x since a0 = 1 and a1 = a.

26. a. Solution USING THE PROPERTIES OF LOGARITHMS Example 4
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. a.
Solution
Product property

27. b. Solution USING THE PROPERTIES OF LOGARITHMS Example 4
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. b.
Solution
Quotient property

28. c. Solution USING THE PROPERTIES OF LOGARITHMS Example 4
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. c.
Solution
Power property

29. Use parentheses to avoid errors.
USING THE PROPERTIES OF LOGARITHMS
Example 4
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. d.
Use parentheses to avoid errors.
Solution
Be careful with signs.

30. e. Solution USING THE PROPERTIES OF LOGARITHMS Example 4
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. e.
Solution
Power property

31. f. USING THE PROPERTIES OF LOGARITHMS Example 4
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. f.
Power property
Product and quotient properties

32. f. Solution USING THE PROPERTIES OF LOGARITHMS Example 4
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1. f.
Solution
Power property
Distributive property

33. a. Solution USING THE PROPERTIES OF LOGARITHMS Example 5
Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. a.
Solution
Product and quotient properties

34. b. Solution USING THE PROPERTIES OF LOGARITHMS Example 5
Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. b.
Solution
Power property
Quotient property

35. c. Solution USING THE PROPERTIES OF LOGARITHMS Example 5
Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. c.
Solution
Power properties

36. c. Solution USING THE PROPERTIES OF LOGARITHMS Example 5
Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. c.
Solution
Product and quotient properties
Rules for exponents

37. c. Solution USING THE PROPERTIES OF LOGARITHMS Example 5
Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1and b ≠ 1. c.
Solution
Rules for exponents
Definition of a1/n

38. Caution There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 5(a),
log3(x + 2) was not written as log3 x + log Remember, log3 x + log3 2 = log3(x • 2).
The distributive property does not apply in a situation like this because log3 (x + y) is one term; “log” is a function name, not a factor.

39. USING THE PROPERTIES OF LOGARITHMS WITH NUMERICAL VALUES
Example 6
Assume that log10 2 = Find each logarithm. a.
Solution b.

40. Theorem on Inverses
For a > 0, a ≠ 1:

41. Theorem on Inverses By the results of this theorem,
The second statement in the theorem will be useful in Sections 4.5 and 4.6 when we solve other logarithmic and exponential equations.