BackExponential Functions in Calculus: Properties, Laws, and Applications
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Exponential Functions
Definition and Basic Properties
An exponential function is a function of the form , where a is a positive real number and . The number a is called the base of the exponential function, and x is the exponent.
If : The function is constant: for all .
Domain: The domain of is .
Range: The range of is .
Properties and Laws of Exponential Functions
Exponential functions have several important properties and laws that govern their behavior:
Increasing/Decreasing: The function is increasing if , and decreasing if .
Identity Property:
Product of Powers:
Quotient of Powers:
Power of a Power:
Product of Different Bases:
Special Point: The point is always on the graph of because .
Graphical Behavior
The graph of shows distinct behaviors depending on the value of the base a:
Base | Graph Behavior |
|---|---|
Increasing function; rises rapidly as increases. | |
Decreasing function; falls rapidly as increases. |
Solving Exponential Equations
To solve equations involving exponential functions, use the properties above to isolate the variable:
If and , , then .
Example 1: Solve for .
Rewrite $27, and $9:
, so
Therefore,
Example 2: Solve for .
Express both sides with base $2$:
,
So,
Set exponents equal:
Solve for as needed.
Applications: Population Growth
Exponential functions are widely used to model population growth, radioactive decay, and other phenomena where change occurs at a constant percentage rate.
Example: A bacterial culture starts with 500 bacteria and increases by 30% every hour.
(a) The population after hours is given by
(b) To find the population after 1, 2, 3, 4, and 5 hours, substitute into the formula.
Natural Exponential Function
Definition and Properties
If the base is the special number e (), the function is called the natural exponential function. This function is always increasing because .
Domain:
Range:
Properties of the Natural Exponential Function
Property | Equation |
|---|---|
Identity | |
Product of Powers | |
Quotient of Powers | |
Power of a Power |
Solving Equations with the Natural Exponential Function
Example 4: Solve for .
Rewrite
Set exponents equal:
Solve:
Example 5: Solve for .
Since , set
Summary Table: Exponential Function Properties
Property | General Exponential () | Natural Exponential () |
|---|---|---|
Identity | ||
Product of Powers | ||
Quotient of Powers | ||
Power of a Power |
Practice Problems
HW: #13, 15-19, 21, 23, 27
