exponentiation

Examples of exponentiation in the following topics:

• Exponentials With Complex Arguments: Euler's Formula

• Exponential Growth and Decay

  • Exponential decay occurs in the same way, providing the growth rate is negative.
  • If $\tau > 0$ and $b > 1$, then $x$ has exponential growth.
  • If $\tau<0$ and $b > 1$, or $\tau > 0$ and $0 < b < 1$, then $x$ has exponential decay.
  • This graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth.
  • Apply the exponential growth and decay formulas to real world examples

• The Natural Exponential Function: Differentiation and Integration

  • Now that we have derived a specific case, let us extend things to the general case of exponential function.
  • Here we consider integration of natural exponential function.
  • Note that the exponential function $y = e^{x}$ is defined as the inverse of $\ln(x)$.

• The Exponential Distribution

  • The exponential distribution is a family of continuous probability distributions.
  • The exponential distribution is often concerned with the amount of time until some specific event occurs.
  • Another important property of the exponential distribution is that it is memoryless.
  • The exponential distribution describes the time for a continuous process to change state.
  • Reliability engineering also makes extensive use of the exponential distribution.

• 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.

• Simplifying Exponential Expressions

  • Multiplying exponential expressions with the same base ($a^m \cdot a^n = a^{m+n}$)
  • In terms of conducting operations, exponential expressions that contain variables are treated just as though they are composed of integers.
  • Applying the rule for dividing exponential expressions with the same base, we have:
  • To simplify the first part of the expression, apply the rule for multiplying two exponential expressions with the same base:

• 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.

• Derivatives of Exponential Functions

  • The derivative of the exponential function is equal to the value of the function.
  • The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative.
  • If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
  • Graph of the exponential function illustrating that its derivative is equal to the value of the function.

• Converting between Exponential and Logarithmic Equations

  • Logarithmic equations can be written as exponential equations and vice versa.
  • The logarithmic equation $\log_b(x)=c$ corresponds to the exponential equation $b^{c}=x$.
  • It might be more familiar if we convert the equation to exponential form giving us:
  • If we write the logarithmic equation as an exponential equation we obtain:
  • We can use this fact to solve such exponential equations as follows:

• Exponential and Logarithmic Functions

  • Both exponential and logarithmic functions are widely used in scientific and engineering applications.
  • The exponential function is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.
  • The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.
  • The exponential function $e^x$ can be characterized in a variety of equivalent ways.
  • The derivative (or slope of a tangential line) of the exponential function is equal to the value of the function.