exponential function

Examples of exponential function in the following topics:

• Graphs of Exponential Functions, Base e

  • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
  • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$ 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.

• Introduction to Exponential and Logarithmic Functions

  • 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 $f(x)=e^x$.

• Basics of Graphing Exponential Functions

  • The exponential function $y=b^x$ 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 $y=b^x$ where $b$ is a positive number.
  • This is true of the graph of all exponential functions of the form $y=b^x$ for $x>1$.
  • This is true of the graph of all exponential functions of the form $y=b^x$ for $0

• Problem-Solving

  • 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 $y=2^x$.

• Population Growth

  • 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.

• Limited Growth

  • 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.

• Restricting Domains to Find Inverses

  • This is not true of the function $f(x)=x^2$.
  • 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.

• Exponential Decay

  • 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.

• Logarithmic Functions

  • Logarithmic equations can be written as exponential equations and vice versa.
  • The logarithmic equation $log_b(x)=c$ corresponds to the exponential equation $b^{c}=x$.
  • As an example, the logarithmic equation $log{_2}16=4$ corresponds to the exponential equation $2^4=16$.
  • 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.

• Exponentials With Complex Arguments: Euler's Formula