College Algebra Corequisite
Module 11:
Exponential and Logarithmic Functions
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Affirmations
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Exponential Functions
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Learning Goals
Deepen your understanding and form connections within these skills:
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Exponential Function Definition
An exponential function has the general form:
where:
• a is a non-zero constant
• b is a positive constant not equal to 1
• b is called the base and x is the exponent
Key Characteristics:
• Domain is (-∞, ∞)
• Range is (0, ∞)
• Horizontal asymptote is y = 0
• y-intercept is (0, a)
• Increasing if b > 1, decreasing if � 0 < b < 1
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Try It!
Determine which of the following represent exponential functions. Explain.
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2.
3.
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Exponential Growth
When b>1, we have exponential growth:
Key features:
Example:
doubles every time x increases by 1
Applications: compound interest, population growth
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Exponential Growth Example
A population starts with 100 bacteria and doubles every hour. Write and evaluate an exponential function for the population after:
Solution:
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Exponential Decay
When 0<b<1, we have exponential decay:
Key features:
Example:
halves every time x increases by 1
Applications: radioactive decay, depreciation
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Exponential Decay Example
A radioactive substance has an initial amount of 80 grams and decays by 25% each day. Find the amount remaining after:
Solution:
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Try It!
Classify each function as exponential growth, exponential decay, or neither.
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2.
3.
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Graphing Exponential Functions
To graph f(x) = b^x:
1. Make a table of points using integer values of x
2. Plot the points, including the y-intercept (0,1)
3. Draw a smooth curve through the points
4. Show the horizontal asymptote y = 0
Example: Graph f(x) = 2^x
The graph passes through (0,1), increases rapidly, approaching y=0 as x → -∞.
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Writing Exponential Functions
Given two points (x₁, y₁) and (x₂, y₂), find constants a and b:
If (0, a) is given:
1) a is the y-intercept
2) Substitute the other point to solve for b
Example: Write an exponential function with y-intercept (0,3) passing through (2,12).
f(x) = 3b^x
12 = 3b^2
b^2 = 4
b = 2
f(x) = 3(2)^x
If y-intercept is not known:
1) Substitute both points into f(x) = ab^x
2) Use logarithms to solve the system for a and b
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Try It!
1. Find an exponential function f(x) = abx passing through (1,6) and (3,1).
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Transformations of Exponential Graphs
Shift Vertically: f(x) = b^x + k
• Shifts the graph up k units if k > 0
• Shifts the graph down k units if k < 0
• The asymptote becomes y = k
Shift Horizontally: f(x) = b^(x-h)
• Shifts left h units if h > 0
• Shifts right h units if h < 0
• y-intercept becomes (h, 1)
Reflect:
• f(x) = -b^x reflects the graph over the x-axis
• f(x) = b^(-x) reflects the graph over the y-axis
Stretch/Compress Vertically: f(x) = ab^x
• Stretches by factor a if |a| > 1
• Compresses by factor 1/|a| if 0 < |a| < 1
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Try It!
1. Graph f(x) = 3x+2-1 and state its domain, range, and asymptote.
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General Exponential Function
Combining all transformations:
f(x) = a·bx-h + k
a: vertical stretch/compress factor and reflection over x-axis
b: base (growth/decay factor)
h: horizontal shift
k: vertical shift
Example: f(x) = -2(4)x+1 - 3
• Compresses vertically by 1/2 and reflects over x-axis
• Base 4 growth factor
• Shifts left 1 unit
• Shifts down 3 units
• Horizontal asymptote at y = -3
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Try It!
Describe each transformation of the parent function f(x) = 2x.
1. g(x) = 2x-3 + 1
2. h(x) = -3(2)x
3. k(x) = 2-x - 4
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Key Takeaways
• Exponential functions model growth or decay at a constant percent rate
• f(x) = abx defines an exponential with base b and vertical shift a
• The graph is increasing if b > 1 and decreasing if 0 < b < 1
• The graph has a horizontal asymptote at y = 0 and domain (-∞, ∞)
• Transformations include shifts, stretches/compressions, and reflections
• The general transformed exponential is f(x) = a·bx-h + k
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Applications
Exponential functions have many real-world applications:
• Population growth or decline
• Radioactive decay
• Compound interest
• Earthquake magnitude
• Equipment depreciation
• Medication elimination
• Cooling and heating
• Spread of news/info
Understanding the behavior of exponentials allows us to model and analyze these various phenomena.
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Applications of Exponential Functions
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Learning Goals
1 Calculate the values of exponential functions, especially those using the base 𝑒, and understand their equations
2 Use compound interest formulas to work out how investments or loans grow over time in real-life financial situations
3 Find an exponential function that models continuous growth or decay
Deepen your understanding and form connections within these skills:
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Applications of Exponential Functions
- Exponential functions are powerful tools for modeling real-world situations
- They can model population growth, radioactive decay, finance, and more
- Exponential functions have the form f(x) = abx, where:
- a is the initial or starting value
- b is the growth factor or multiplier per unit x
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Bacteria Growth Example
[Include table showing bacteria growth]
- Bacteria can reproduce very quickly through binary fission
- In this example, one bacterial cell splits into two each hour
- The exponential growth leads to over 1000 cells in just 10 hours!
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Exponential Growth Function
f(x) = abx
where:
- a is the initial or starting value
- b is the growth factor or multiplier per unit x
- If b > 1, the function represents exponential growth
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Try It!
1. The population of a small town was 5,000 in the year 2010 and has been growing at an annual rate of 4%. Write an exponential function representing this situation. To the nearest whole number, what will the population be in 2025?
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Compound Interest
- Interest earned on both the principal and accumulated interest
- Nominal rate / Annual Percentage Rate (APR): Yearly rate earned by an account
- Compounding: Interest earned on interest
- The more frequently interest compounds, the more interest is earned
Compound Interest Formula
A(t) = P(1 + r/n)^(nt)
where:
- A(t) is the account value at time t
- P is the principal or starting amount
- r is the APR expressed as a decimal
- n is the number of compounding periods per year
- t is the time in years
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Try It!
1. You deposit $6,500 into an account that earns 3.2% interest compounded monthly. How much will you have after 5 years?
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Evaluating Exponential Functions with base e
[Include table showing values approaching e]
- As compounding frequency increases, the account value approaches a limit
- That limit is represented by the irrational number e ≈ 2.718282
- e was discovered by Leonhard Euler and has many applications
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Continuous Growth and Decay
The exponential function with base e models continuous growth or decay:
A(t) = Pert
where:
- P is the initial value
- r is the growth rate (r > 0) or decay rate (r < 0)
- t is the time
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Try It!
1. Suppose a car depreciates continuously at a rate of 5% per year. If you buy the car for $30,000, what will it be worth after 4 years?
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Logarithmic Functions
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Learning Goals
Deepen your understanding and form connections within these skills:
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Converting Between Logarithmic and Exponential Form
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Logarithmic Function Definition
For x > 0, b > 0, b ≠ 1
y = logb(x) is equal to bʸ = x
where:
logb(x) is read as "the logarithm with base b of x" or "log base b of x"
y is the exponent to which b must be raised to get x
If no base is indicated, base 10 is assumed
Domain: (0, ∞)
Range: (-∞, ∞)
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Try It!
1. Write the equation 7³ = 343 in logarithmic form.
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Converting from Logarithmic to Exponential Form
Given: y = logb(x), convert it to exponential form
Steps:
1. Identify b, y, and x
2. Rewrite logb(x) = y as bʸ = x
Example:
Convert log₆(√6) = 1/2 to exponential form.
Here, b = 6, y = 1/2, x = √6
log₆(√6) = 1/2 is equal to 6¹ᐟ² = √6
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Try It!
1. Convert log₃(1/9) = -2 to exponential form.
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Converting from Exponential to Logarithmic Form
Given: x = bʸ, convert it to logarithmic form
Steps:
1. Identify b, y, and x
2. Rewrite bʸ = x as y = logb(x)
Example:
Convert 10⁻³ = 1/1000 to logarithmic form.
Here, b = 10, y = -3, x = 1/1000
10⁻³ = 1/1000 is equal to log₁₀(1/1000) = -3
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Try It!
1. Convert 5⁴ = 625 to logarithmic form.
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Evaluating Logarithms
To evaluate logb(x) mentally:
1. Rewrite x as a power of b: bʸ = x
2. Identify y by asking, "To what exponent should b be raised to get x?"
Example:
Solve y = log₄(64) without a calculator.
Rewrite: 4ʸ = 64
4³ = 64, so log₄(64) = 3
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Try It!
1. Evaluate log₈(1/512) without using a calculator.
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Common Logarithms
A common logarithm is a logarithm with base 10
We write log₁₀(x) simply as log(x)
For x > 0, y = log(x) is equivalent to 10ʸ = x
Example: Evaluate log(100)
100 = 10², so log(100) = 2
To evaluate log(x) when x is not a power of 10, use a calculator
Calculator Steps:
1. Press [LOG]
2. Enter the x value, followed by [ ) ]
3. Press [ENTER]
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Try It!
Evaluate log(79) using a calculator. Round to 4 decimal places.
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2.
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Natural Logarithms
A natural logarithm is a logarithm with base e
We write logₑ(x) as ln(x)
For x > 0, y = ln(x) is equivalent to eʸ = x
e ≈ 2.71828
Calculator Steps for ln(x):
1. Press [LN]
2. Enter the x value, followed by [ ) ]
3. Press [ENTER]
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Try It!
1. Evaluate ln(0.5) using a calculator. Round to 4 decimal places.
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Natural Logarithms (continued)
Properties of natural logarithm:
- ln(1) = 0
- ln(eˣ) = x for all x
- eˡⁿ⁽ˣ⁾ = x for x > 0
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Try It!
1. Evaluate ln(e³).
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Logarithmic Function Graphs and Characteristics
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Learning Goals
Deepen your understanding and form connections within these skills:
Identify
Identify the domain of a logarithmic function
x.1
Graph
Graph logarithmic functions
x.2
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Domain of Logarithmic Functions
- Logarithmic functions are only defined for positive real numbers
- The argument (input) of the logarithm must be greater than zero
- This results in a vertical asymptote at x = 0
- The domain is (0, ∞) and the range is (-∞, ∞)
Example:
Find the domain of f(x) = log₂(x + 3)
x + 3 > 0
x > -3
The domain is (-3, ∞)
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Steps to Find the Domain of a Logarithmic Function
1. Set up an inequality showing the argument greater than zero
2. Solve for x
3. Write the domain in interval notation
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Try It!
1. What is the domain of f(x) = log(5 - 2x)?
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Graphing Logarithmic Functions
- Logarithmic functions are the inverses of exponential functions
- Their graphs are reflections across the line y = x
- The parent function is f(x) = logb(x)
Key characteristics of the parent function f(x) = logb(x):
- One-to-one function
- Vertical asymptote: x = 0
- Domain: (0, ∞)
- Range: (-∞, ∞)
- x-intercept: (1, 0) and key point (b, 1)
- No y-intercept
- Increasing if b > 1, decreasing if 0 < b < 1
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Steps to Graph a Logarithmic Function f(x) = logb(x)
1. Draw and label the vertical asymptote, x = 0
2. Plot the x-intercept, (1, 0)
3. Plot the key point (b, 1)
4. Draw a smooth curve through the points
5. State the domain (0, ∞), range (-∞, ∞), and vertical asymptote x = 0
Example:
Graph f(x) = log₅(x)
[Include graph with key points (1/5, -1), (1, 0), (5, 1), vertical asymptote at x = 0]
The domain is (0, ∞), the range is (-∞, ∞), and the vertical asymptote is x = 0.
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Horizontal Shifts of Logarithmic Functions
For f(x) = logb(x + c):
- c > 0: shift left c units
- c < 0: shift right c units
- Vertical asymptote: x = -c
- Domain: (-c, ∞)
- Range: (-∞, ∞)
Steps:
1. Identify the horizontal shift
2. Draw the vertical asymptote x = -c
3. Find new coordinates by subtracting c from parent function x-coordinates
4. Label points, domain (c, ∞), range (-∞, ∞), vertical asymptote x = -c
Example:
Graph f(x) = log₃(x - 2)
[Include graph of parent function and f(x) with key points, asymptote at x = 2]
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Vertical Shifts of Logarithmic Functions
For f(x) = logb(x) + d:
- d > 0: shift up d units
- d < 0: shift down d units
- Vertical asymptote, domain, range unchanged
Steps:
1. Identify the vertical shift
2. Draw vertical asymptote x = 0
3. Find new coordinates by adding d to parent function y-coordinates
4. Label points, domain (0, ∞), range (-∞, ∞), vertical asymptote x = 0
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Try It!
1. Sketch f(x) = log(x) - 1
Solution:
[Graph with parent function, f(x) shifted down 1 unit, asymptote at x = 0, points (1, -1), (10, 1)]
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Stretches and Compressions of Logarithmic Functions
For f(x) = a·logb(x), a > 0:
- |a| > 1: stretch vertically by factor of a
- |a| < 1: compress vertically by factor of a
- Vertical asymptote, x-intercept (1,0), domain, range unchanged
Steps:
1. Identify stretch/compression factor a
2. Draw vertical asymptote x = 0
3. Multiply parent function y-coordinates by a for new coordinates
4. Label points, domain (0, ∞), range (-∞, ∞), vertical asymptote x = 0
Example:
Graph f(x) = (1/2)log₄(x)
[Graph with parent function and f(x) compressed vertically by 1/2, asymptote at x = 0]
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Reflections of Logarithmic Functions
f(x) = -logb(x):
- Reflects parent function about x-axis
- Domain, range, vertical asymptote x = 0 unchanged
f(x) = logb(-x):
- Reflects parent function about y-axis
- Domain: (-∞, 0)
- Range, vertical asymptote x = 0 unchanged
Steps:
1. Draw vertical asymptote x = 0
2. Reflect across x-axis for -logb(x) or y-axis for logb(-x)
3. Plot x-intercept (1,0) or (-1,0) respectively
4. Draw smooth curve
5. Label domain, range (-∞, ∞), vertical asymptote
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Try It!
1. Graph f(x) = -log(x)
Solution:
[Graph with parent log(x), f(x) reflected across x-axis, asymptote at x = 0, point (1,0)]
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Closing Slide
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Next Steps…..
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Questions…..
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Anything else….
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