Examples of exponential function in the following topics:
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- The function f(x)=ex is a basic exponential function with some very interesting properties.
- The basic exponential function, sometimes referred to as the exponential function, is f(x)=ex where e is the number (approximately 2.718281828) described previously.
- The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.
- If the change is positive, this is called exponential growth and if it is negative, it is called exponential decay.
- For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function.
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- Logarithmic functions and exponential functions are inverses of each other.
- The inverse of an exponential function is a logarithmic function and vice versa.
- That is, the two functions undo each other.
- In the following graph you can see an exponential function in red and its inverse, a logarithmic function, in blue.
- The natural logarithm is the inverse of the exponential function f(x)=ex.
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- The exponential function y=bx where b>0 is a function that will remain proportional to its original value when it grows or decays.
- At the most basic level, an exponential function is a function in which the variable appears in the exponent.
- The most basic exponential function is a function of the form y=bx where b is a positive number.
- This is true of the graph of all exponential functions of the form y=bx for x>1.
- This is true of the graph of all exponential functions of the form y=bx for $0
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- Graphically solving problems with exponential functions allows visualization of sometimes complicated interrelationships.
- The exponential function has numerous applications.
- Biology, Economics, and Finance are only just a few of the applications that use exponential functions.
- In general, a real world application that can be modeled with an exponential function is one that will continuously increase at an increasing rate based on the current value.
- Graph of an exponential function with the equation y=2x.
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- Population can fluctuate positively or negatively and can be modeled using an exponential function.
- Population growth can be modeled by an exponential equation.
- The rate r by which the population is growing is itself a function of four variables.
- If the current rates of births and deaths hold, the world population growth can be modeled using an exponential function.
- The graph below shows an exponential model for the growth of the world population.
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- Exponential functions can be used to model growth and decay.
- Even if we account for varying rates of growth, the idea that human population can be modeled strictly with an exponential function is misguided.
- Exponential functions are ever-increasing so saying that an exponential function models population growth exactly means that the human population will grow without bound.
- Graphically, the logistic function resembles an exponential function followed by a logarithmic function that approaches a horizontal asymptote.
- Below is the graph of a logistic function.
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- This is not true of the function f(x)=x2.
- The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
- If so, what is the value of the function when x=0?
- No, the function has no defined value for x=0.
- Rewriting as an exponential equation gives: 10n=0, which is impossible since no such real number n exists.
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- Exponential decay is the result of a function that decreases in proportion to its current value.
- Just as it is possible for a variable to grow exponentially as a function of another, so can the a variable decrease exponentially.
- Exponential rate of change can be modeled algebraically by the following formula:
- The exponential decay of the substance is a time-dependent decline and a prime example of exponential decay.
- Below is a graph highlighting exponential decay of a radioactive substance.
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- Logarithmic equations can be written as exponential equations and vice versa.
- The logarithmic equation logb(x)=c corresponds to the exponential equation bc=x.
- As an example, the logarithmic equation log216=4 corresponds to the exponential equation 24=16.
- It might be more familiar if we convert the equation to exponential form giving us:
- The explanation of the previous example reveals the inverse of the logarithmic operation: exponentiation.
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