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Logarithmic Functions
- Solve the equations below.
𝑥𝑥 2 = 4 𝑥𝑥 2 = 5
- Were you able solve both equations above? If so, was one of the equations easier to solve than the other? Why?
- Solve the equations below.
2 𝑥𝑥^ = 32 2 𝑥𝑥^ = 30
- Were you able solve both equations above? If so, was one of the equations easier to solve than the other? Why?
Consider the function 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥. If we attempt to find the inverse function (is f one-to-one?), we see:
𝑓𝑓(𝑥𝑥) = 2𝑥𝑥 𝑦𝑦 = 2𝑥𝑥 Now switching x and y : 𝑥𝑥 = 2𝑦𝑦
From here, how can we solve for y? The answer is – we can’t. We need to define a new function, a logarithmic function , which allows us to “undo” an exponential function.
If 𝑥𝑥 and 𝑏𝑏 are positive real numbers such that 𝑏𝑏 ≠ 1 , then 𝑦𝑦 = log𝑏𝑏 𝑥𝑥 is called the logarithmic function base 𝒃𝒃, where 𝑦𝑦 = log𝑏𝑏 𝑥𝑥 ↔ 𝑥𝑥 = 𝑏𝑏 𝑦𝑦
Notes :
- The value 𝑦𝑦 is the exponent to which 𝑏𝑏 must be raised to obtain 𝑥𝑥.
- The value of 𝑦𝑦 is called the logarithm , 𝑏𝑏 is called the base , and 𝑥𝑥 is called the argument.
- The equations 𝑦𝑦 = log𝑏𝑏 𝑥𝑥 and 𝑥𝑥 = 𝑏𝑏 𝑦𝑦^ both define the same relationship between 𝑥𝑥 and 𝑦𝑦. The expression 𝑦𝑦 = log𝑏𝑏 𝑥𝑥 is called the logarithmic form and 𝑥𝑥 = 𝑏𝑏 𝑦𝑦^ is called the exponential form.
In addition to the general form of a logarithm as shown above, we have special notation for two of the logarithms.
The natural logarithm function is denoted 𝑓𝑓(𝑥𝑥) = ln 𝑥𝑥. The common logarithm function is denoted 𝑓𝑓(𝑥𝑥) = log 𝑥𝑥.
- Write each equation in exponential form:
a. log 3 9 = 2
b. log 1000 = 3
- Write each equation in logarithmic form:
a. 52 = 25
b. 100 −2^ = 1 10000
**Make sure to say:
“log base b of 1” “log base b of b ” “ b to the power log base b of x ” “log base b of b to the x ”
10
8 6 4 2 2 4 6 8
10
10 5 5 10
f ( x ) = 2 x
10
8 6 4 2 2 4 6 8
10
10 5 5 10
g ( x ) = 3 x
10
8 6 4 2 2 4 6 8
10
10 5 5 10
h ( x ) = 10 x
10
8 6 4 2 2 4 6 8
10
10 5 5 10
j ( x ) = e x
Basic Logarithmic Properties For 𝑏𝑏 > 0, 𝑏𝑏 ≠ 1:
log𝑏𝑏 1 = log𝑏𝑏 𝑏𝑏 = 𝑏𝑏 log𝑏𝑏^ 𝑥𝑥^ = log𝑏𝑏 𝑏𝑏 𝑥𝑥^ =
- Recall the graphs of 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥, 𝑔𝑔(𝑥𝑥) = 3𝑥𝑥, ℎ(𝑥𝑥) = 10𝑥𝑥, and 𝑗𝑗(𝑥𝑥) = 𝑒𝑒 𝑥𝑥. Use them to plot the inverses 𝑓𝑓 −1(𝑥𝑥) = log 2 𝑥𝑥, 𝑔𝑔 −1(𝑥𝑥) = log 3 𝑥𝑥, ℎ −1(𝑥𝑥) = log 𝑥𝑥, and 𝑗𝑗 −1(𝑥𝑥) = ln 𝑥𝑥.
- Recalling that 𝑦𝑦 = log𝑏𝑏 𝑥𝑥 ↔ 𝑥𝑥 = 𝑏𝑏 𝑦𝑦^ and referencing the graphs you just created, are there any values of x that we cannot use? In other words, what is the domain of a logarithm of the form 𝑦𝑦 = log𝑏𝑏 𝑥𝑥?
- What is the range of a logarithm of the form 𝑦𝑦 = log𝑏𝑏 𝑥𝑥?
- What is the connection between the domain and range of the exponential functions versus the logarithmic functions?
Recall the properties of the graphs of exponential functions:
The graph of an exponential function 𝑓𝑓(𝑥𝑥) = 𝑏𝑏 𝑥𝑥^ has the following properties.
- If 𝑏𝑏 > 1, 𝑓𝑓 is an increasing exponential function ( exponential growth function).
If 0 < 𝑏𝑏 < 1, 𝑓𝑓 is a decreasing exponential function ( exponential decay function).
- The domain is the set of all real numbers, (−∞, ∞).
- The range is (0, ∞).
- The line 𝑦𝑦 = 0 is a horizontal asymptote.
- The function passes through the point (0, 1) because 𝑓𝑓(0) = 𝑏𝑏 0 = 1. We will unofficially call this the “ pivot point .”
Compare to the properties of the graphs of logarithmic functions:
The graph of a logarithmic function 𝑓𝑓(𝑥𝑥) = log𝑏𝑏 𝑥𝑥 has the following properties.
- If 𝑏𝑏 > 1, 𝑓𝑓 is an increasing logarithmic function ( logarithmic growth function). If 0 < 𝑏𝑏 < 1, 𝑓𝑓 is a decreasing logarithmic function ( logarithmic decay function).
- The domain is (0, ∞).
- The range is the set of all real numbers, (−∞, ∞).
- The line 𝑥𝑥 = 0 is a vertical asymptote.
- The function passes through the point (1, 0) because 𝑓𝑓(0) = log𝑏𝑏 1 = 0. We will unofficially call this the “ pivot point .”
8 6 4 2 2 4 6 8
5 5
- Graph 𝑓𝑓(𝑥𝑥) = log 3 𝑥𝑥 and 𝑔𝑔(𝑥𝑥) = log 3 (𝑥𝑥 − 1) + 3 on the same set of axes. List the transformations you would apply to 𝑓𝑓(𝑥𝑥) to get 𝑔𝑔(𝑥𝑥) and label the intercepts, pivot point, and asymptote.
Transformations:
- In number 9, you should have determined that the domain of a function of the form 𝑓𝑓(𝑥𝑥) = log𝑏𝑏 𝑥𝑥 is
(0, ∞). If there are horizontal transformations, the domain will change. To determine the domain of any logarithmic function, the argument of the logarithm must be positive. Use this information to determine the domain of each of the following functions. Write your answer in interval notation.
a. 𝑓𝑓(𝑥𝑥) = log 5 (𝑥𝑥 + 2)
b. 𝑝𝑝(𝑥𝑥) = ln(6 − 𝑥𝑥)
c. 𝑓𝑓(𝑥𝑥) = log 4 (3𝑥𝑥 − 7) + 4
d. 𝑞𝑞(𝑥𝑥) = log 2 (𝑥𝑥 2 − 5 𝑥𝑥 − 14)
e. 𝑟𝑟(𝑥𝑥) =
log 2 �𝑥𝑥 2 −5𝑥𝑥−14�+√2𝑥𝑥+ 𝑥𝑥−
