exponential growth

(noun)

The growth in the value of a quantity, in which the rate of growth is proportional to the instantaneous value of the quantity; for example, when the value has doubled, the rate of increase will also have doubled. The rate may be positive or negative. If negative, it is also known as exponential decay.

Related Terms

Examples of exponential growth in the following topics:

  • Geometric Sequences

    • Greater than $1$, there will be exponential growth towards positive infinity.
    • Between $-1$ and $1$ but not $0$, there will be exponential decay toward $0$.
    • Less than $-1$, for the absolute values there is exponential growth toward positive and negative infinity (due to the alternating sign).
    • Geometric sequences (with common ratio not equal to $-1$, $1$ or $0$) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as $4, 15, 26, 37, 48, \cdots$ (with common difference $11$).
    • Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.

  • Limited Growth

    • A realistic model of exponential growth must dampen when approaching a certain value.
    • Exponential functions can be used to model growth and decay.
    • Exponential functions are ever-increasing so saying that an exponential function models population growth exactly means that the human population will grow without bound.
    • This is the idea behind limited growth, that a population may grow exponentially but only to a point at which time the growth will taper off.
    • Thus, the model of population growth among sheep will no longer be exponential.

  • 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.
    • 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.
    • Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as $y=Ae^{kt}$ where $A$ is the number of bacteria present initially (at time $t=0$) and $k$ is a constant called the growth constant.

  • Problem-Solving

    • The exponential function has numerous applications.
    • Economic growth is expressed in percentage terms, implying exponential growth.
    • Compound interest at a constant interest rate provides exponential growth of the capital.
    • Think about a microorganism's growth.
    • In what other circumstances would you see exponential growth?

  • 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.
    • It is the Population Growth Rate ($PGR$).
    • 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.

  • Solving General Problems with Logarithms and Exponents

    • Logarithms are useful for solving equations that require an exponential term, like population growth.
    • This population growth graph shows that it grows exponentially with time.

  • Basics of Graphing Exponential Functions

    • The most basic exponential function is a function of the form $y=b^x$ where $b$ is a positive number.
    • This is called exponential growth.
    • This is called exponential decay.
    • 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

  • Natural Logarithms

    • Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$, also arises in naturally in many contexts.
    • For example, the doubling time for a population which is growing exponentially is usually given as ${\ln 2 \over k}$ where $k$ is the growth rate, and the half-life of a radioactive substance is usually given as ${\ln 2 \over \lambda}$ where $\lambda$ is the decay constant.

  • Introduction to Exponents

    • Exponential form, written $b^n$, represents multiplying the base $b$ times itself $n$ times.
    • Exponentiation is a mathematical operation that represents repeated multiplication.
    • Exponentiation is used frequently in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public key cryptography.
    • Now that we understand the basic idea, let's practice simplifying some exponential expressions.
    • Let's look at an exponential expression with 2 as the base and 3 as the exponent:

  • Exponentials With Complex Arguments: Euler's Formula