7- 3 Logarithmic Functions 505 How many times would you have to double $1 before you had $8? You could use an exponential equation to model this situation. 1 (2 x ) = 8. You may be able to solve this equation by using mental math if you know that 2 3 = 8. So you would have to double the dollar 3 times to have $8. How many times would you have to double $1 to have $512? You could solve this problem if you could solve 2 x = 8 by using an inverse operation that undoes raising a base to an exponent. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value. You can write an exponential equation as a logarithmic equation and vice versa. Exponential Equation b > 0, b ≠ 1 Logarithmic Equation 1 EXAMPLE Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form a. 2 6 = 64 log 2 64 = 6 b. 4 1 = 4 log 4 4 = 1 c. 5 0 = 1 log 5 1 = 0 d. 5 -2 = 0.04 log 5 0.04 = -2 e. 3 x = 81 log 3 81 = x The base of the exponent becomes the base of the logarithm. The exponent is the logarithm. Any nonzero base to the 0 power is 1. An exponent (or log) can be negative. The log (and the exponent) can be a variable. Write each exponential equation in logarithmic form. 1a. 9 2 = 81 1b. 3 3 = 27 1c. x 0 = 1 (x ≠ 0) 7-3 Logarithmic Functions Objectives Write equivalent forms for exponential and logarithmic functions. Write, evaluate, and graph logarithmic functions. Vocabulary logarithm common logarithm logarithmic function Why learn this? A logarithmic scale is used to measure the acidity, or pH, of water. (See Example 5.) Read log b a = x, as “the log base b of a is x.” Notice that the log is the exponent. 505 TEKS 2A.11.A Exponential and logarithmic functions: develop the definition of logarithms by … describing the relationship between exponential functions and their inverses. Also 2A.4.A, 2A.11.C
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7- 3 Logarithmic Functions 505
How many times would you have to double $1 before you had $8? You could use an exponential equation to model this situation. 1 ( 2 x ) = 8. You may be able to solve this equation by using mental math if you know that 2 3 = 8. So you would have to double the dollar 3 times to have $8.
How many times would you have to double $1 to have $512? You could solve this problem if you could solve 2 x = 8 by using an inverse operation that undoes raising a base to an exponent. This operation is called finding the logarithm. A logarithm is the exponent to which a specified base is raised to obtain a given value.
You can write an exponential equation as a logarithmic equation and vice versa.
Exponential Equation
b > 0, b ≠ 1
Logarithmic Equation
1E X A M P L E Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
Exponential Equation
Logarithmic Form
a. 2 6 = 64 log 2 64 = 6
b. 4 1 = 4 log 4 4 = 1
c. 5 0 = 1 log 5 1 = 0
d. 5 -2 = 0.04 log 5 0.04 = -2
e. 3 x = 81 log 3 81 = x
The base of the exponent becomes the base of the logarithm.
The exponent is the logarithm.
Any nonzero base to the 0 power is 1.
An exponent (or log) can be negative.
The log (and the exponent) can be a variable.
Write each exponential equation in logarithmic form.
1a. 9 2 = 81 1b. 3 3 = 27 1c. x 0 = 1 (x ≠ 0)
7-3 Logarithmic Functions
ObjectivesWrite equivalent forms for exponential and logarithmic functions.
Write, evaluate, and graph logarithmic functions.
Vocabularylogarithmcommon logarithmlogarithmic function
Why learn this?A logarithmic scale is used to measure the acidity, or pH, of water. (See Example 5.)
Read lo g b a = x, as “the log base b of a is x.” Notice that the log is the exponent.
505
TEKS 2A.11.A Exponential and logarithmic functions: develop the definition of logarithms by … describing the relationship between exponential functions and their inverses.
A logarithm is an exponent, so the rules for exponents also apply to logarithms. You may have noticed the following properties in the last example.
For any base b such that b > 0 and b ≠ 1,
LOGARITHMIC FORM EXPONENTIAL FORM EXAMPLE
Logarithm of Base b
lo g b b = 1 b 1 = b
lo g 10 10 = 1 10 1 = 10
Logarithm of 1
lo g b 1 = 0 b 0 = 1lo g 10 1 = 0 10 0 = 1
Special Properties of Logarithms
A logarithm with base 10 is called a common logarithm . If no base is written for a logarithm, the base is assumed to be 10. For example, log 5 = log 10 5.
You can use mental math to evaluate some logarithms.
3E X A M P L E Evaluating Logarithms by Using Mental Math
Because logarithms are the inverses of exponents, the inverse of an exponential function, such as y = 2 x , is a logarithmic function , such as y = log 2 x.
You may notice that the domain and range of each function are switched.
The domain of y = 2 x is all real numbers (�), and the range is
⎧ ⎨
⎩ y | y > 0
⎫ ⎬
⎭ . The domain of y = log 2 x is
⎧ ⎨
⎩ x | x > 0
⎫ ⎬
⎭ , and the range is all real numbers (�).
4E X A M P L E Graphing Logarithmic Functions
Use the given x-values to graph each function. Then graph its inverse. Describe the domain and range of the inverse function.
A f (x) = 3 x ; x = -2, -1, 0, 1, and 2
Graph f (x) = 3 x by using a table of values.
x -2 -1 0 1 2
f (x) = 3 x 1 _ 9 1 _
3 1 3 9
To graph the inverse, f -1 (x) = log 3 x, reverse each ordered pair.
x 1 _ 9 1 _
3 1 3 9
f -1 (x) = log 3 x -2 -1 0 1 2
The domain of f -1 (x) is ⎧ ⎨
⎩ x | x > 0
⎫ ⎬
⎭ , and the range is �.
B f (x) = 0. 8 x ; x = -3, 0, 1, 4, and 7
Graph f (x) = 0. 8 x by using a table of values. Round the output values to the nearest tenth, if necessary.
x -3 0 1 4 7
f (x) = 0.8 x 2 1 0.8 0.4 0.2
To graph f -1 (x) = log 0.8 x, reverse each ordered pair.
x 2 1 0.8 0.4 0.2
f -1 (x) = log 0.8 x -3 0 1 4 7
The domain of f -1 (x) is ⎧ ⎨
⎩ x | x > 0
⎫ ⎬
⎭ , and the range is �.
4. Use x = -2, -1, 1, 2, and 3 to graph f (x) = ( 3 __ 4 ) x . Then
graph its inverse. Describe the domain and range of the inverse function.
508 Chapter 7 Exponential and Logarithmic Functions
5E X A M P L E Environmental Application
Chemists regularly test rain samples to determine the rain’s acidity, or concentration of hydrogen ions ( H + ) . Acidity is measured in pH, as given by the function pH = - log ⎡ ⎣ H + ⎤ ⎦ , where ⎡ ⎣ H + ⎤ ⎦ represents the hydrogen ion concentration in moles per liter.
Find the pH of rainwater from each location.
A Central New Jersey
The hydrogen ion concentration is 0.0000316 moles per liter.
pH = - log ⎡ ⎣ H + ⎤ ⎦
pH = - log (0.0000316) Substitute the known values in the function.
Use a calculator to find the value of the logarithm in base 10. Press the key.
The rainwater has a pH of about 4.5.
B Central North Dakota
The hydrogen ion concentration is 0.0000009 moles per liter.
pH = - log ⎡ ⎣ H + ⎤ ⎦
pH = - log (0.0000009) Substitute the known values in the function.
Use a calculator to find the value of the logarithm in base 10. Press the key.
The rainwater has a pH of about 6.0.
5. What is the pH of iced tea with a hydrogen ion concentration of 0.000158 moles per liter?
THINK AND DISCUSS1. Explain why log b b is always equal to 1 for b > 0 and b ≠ 1.
2. Explain whether log b a is the same as log a b. Support your answer.
3. GET ORGANIZED Copy and complete the graphic organizer. Use your own words to explain a logarithmic function.