Understanding Exponential Functions
An exponential function is a mathematical function that has a variable as the exponent of a constant number. The general form of an exponential function is with , , and .
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An exponential function is a mathematical function that has a variable as the exponent of a constant number. The general form of an exponential function is with , , and .
Components of exponential functions:
In the function :
Exponential functions have several special properties that distinguish them from other functions:
Exponential Growth Function ():
Exponential Decay Function ():
The following is a visualization of various exponential functions:
Comparison of function values and :
Exponential functions can be transformed in various ways:
The function shifts the graph upward (if ) or downward (if ).
The function shifts the graph to the right (if ) or to the left (if ).
Exponential functions are widely used in daily life:
Living organism populations often follow exponential growth patterns. If the initial population is and the growth rate is per time period, then:
Example: Bacterial population that reproduces every hour at a rate of :
| Time (hours) | ||||||
|---|---|---|---|---|---|---|
| Population |
Radioactive substances decay following exponential functions. If the initial mass is and the half-life is , then:
Investments with compound interest grow exponentially. If the initial capital is , interest rate is per year, and time is :
where is the frequency of interest compounding per year.
An exponential equation is an equation that contains a variable in the exponent. General form:
Method : Equalizing Bases
If , then
Example: Solve
Method : Using Logarithms
To solve , use logarithms:
Determine the value of if
Solve the equation
A city's population is and grows per year. What will the population be after ?
A radioactive substance has a half-life of . If the initial mass is , how much mass remains after ?
, so , therefore