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Exponential and Logarithmic Function Gallery

Lets take a look at the background of the two basic transcendental functions in this gallery. A transcendental function is a mathematical function that cannot be expressed as a finite combination of algebraic operations, such as addition, subtraction, multiplication, division, raising to a power, and extracting a root. (An algebraic function is a function that involves only these operations.) Examples of transcendental functions include the exponential function, the natural logarithm, and the trigonometric functions. (There are others.) (A portion of this introduction appeared in Transcendental Function.) For a trig function gallery click here TRIG GALLERY.

Both of our "basic" functions have a long history and have changed their formulation as more mathematical topics provided added improvements. Changes occurred from varied sources like Egypt, Islamic Golden Age, Persian mathematicians, Archimedes, Greece, India, and through out the 15th-18th centuries. As things evolved the basic functions became intertwined so that we have a pair of functions which play important roles in mathematics and a variety of fields like astronomy, physics, economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, public-key cryptography, radioactive decay, measuring earthquakes, measuring sound decibels, and measuring pH balance. (A portion of this paragraph appeared in Some History.)

The exponential function is a mathematical function denoted by \( f(x) =exp(x)\) or \(e^x\) (where the argument \(x\) is written as an exponent). The functions \(f(x)=b^x\) for positive real numbers \(b\) are also known as exponential functions. The natural base \(e = exp(1) =2.71828…\) is a ubiquitous (Being or seeming to be everywhere at the same time.) mathematical constant called Euler's number. To distinguish it, \(exp(x)= e^x\) is called the exponential function or the natural exponential function. The constant was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest. The exponential function is one of the most important functions in mathematics but the linear function ranks even higher in importance. (Portions for this paragraph came from Exp Function, Exponentiation, more Exp Function, and Math Constant.)

The history of logarithms is the story of a correspondence between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer. The Scottish mathematician John Napier published his discovery of logarithms in 1614. His purpose was to assist in the multiplication of quantities that were then called sines. E. W. Hobson called it "one of the very greatest scientific discoveries that the world has seen". Henry Briggs introduced common or natural (base 10) logarithms, which were easier to use than the original Napierian logarithms. Tables of logarithms were published in many forms over four centuries. The idea of logarithms was also used to construct the slide rule, which became ubiquitous in science and engineering until the 1970s.

Optional Comments: If you haven't seen a slide rule or couldn't find your slide rule, click here Slide Rule 1960. (This is my 1960 slide rule with case. It was a bit dusty.I needed it for Calculus I.) For directions on using a slide rule click here How to use a slide rule.

The first hand held calculator was invented in 1967. But it weighed 55 pounds and cost $2,500. It needed more work. For a bit of history click here Early calculator.

"Resonable" hand held calculators were available in 1972-73. The Texas Instrument's SR-10 cost me $100.00. I got it to use with the first Numerical Analysis course that I taught. Of course the students did not have one. Click here TI SR-10 1973 for a picture. (I dusted this off also.)

The natural logarithm with base \(e \cong 2.71828\) and written \(ln\)), however, continues to be one of the most useful functions in mathematics, with applications to mathematical models throughout the physical and biological sciences. A breakthrough generating the natural logarithm required the assimilation of a new function into standard mathematics. (Portions of these paragraphs came from History of logarithms and Britannica logarithm.)

Previously, we said our basic transcendental functions were intertwined. The function \(\mathbf{y = log_{a}(x)}\) is the inverse of the function \(\mathbf{y = a^{x} }\). In other words,

Inverse display

"whenever these make sense". That is, we have restrictions regarding domains and ranges of our basic functions.

Domain-range display

For more information click here Further details.

The gallery contains two particular equations based on the exponential function. The logistic equation or logistic curve is a common S-shaped curve. The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The logistic equation appears in a wide range of fields like ecology, chemistry, demography, economics, geoscience, probability, sociology, political science, linguistics, and statistics. For more information click here Logistic equation.

The second equation is called the normal density function. "A normal distribution is sometimes informally called a bell curve." "The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P. G. Hoel (1947) Introduction to Mathematical Statistics and A. M. Mood (1950) Introduction to the Theory of Statistics. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known." For more information click here Normal distribution.

Exponential and Logarithmic Gallery

The following is a gallery of demos for illustrating selected families of functions. These figures and animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the change of parameters in expressions for functions. Two file formats, gif and MP4, are used and should run on most systems. Originally, see the NOTES below the gallery of animations, the animations were gif files which had no control for a user. The animations are now MP4 files which allow the user to easily control features like stop and go, restart, and change the animation screen size. The controls are similar to those in You Tube. Click this thumbnail MP4_Features to see an example.

Gallery of Animations

\(\text{General exponential function}\)

\( \mathbf{ \color{red}{h(x)=a^x}}\)
\( \mathbf{ \color{green}{f(x)=a^x+b}}\)
gallery_gen_exp_pic Vary a MP4 Animation

Vary b MP4 Animation

\(\text{Exponential function base e}\)

\( \mathbf{ \color{red}{h(x)=a*e^{x+b}}}\)
\( \mathbf{ \color{green}{f(x)=a*e^{x+b}}+c}\)
gallery_exp_pic Vary a MP4 Animation

Vary b MP4 Animation

Vary c MP4 Animation

\(\text{Natural logarithm function}\)

\( \mathbf{ \color{red}{h(x)=a*ln{(x+b)}}}\)
\( \mathbf{ \color{green}{f(x)=a*ln{(x+b)}}+c}\)
gallery_Natural_logarithm_function_pic Vary a MP4 Animation

Vary b MP4 Animation

Vary c MP4 Animation

\(\text{Negative exponential family}\)

\( \mathbf{ \color{green}{r(x)=}\color{red}{a}\color{green}{*}\color{green}{(1-e^{-\color{red}{b}\color{green}{*x})}}}\)
gallery_neg_exp_pic Vary a MP4 Animation

Vary b MP4 Animation

\(\text{Logistic equation family}\)

\(\mathbf{ \color{green}{r(x) = } \frac{\color{red}{C} } { \color{green}{1 +} \color{red}{A} \color{green}{* e^{- bx} } } }\)
gallery_logistic_pic Vary A MP4 Animation

Vary b MP4 Animation

Vary C MP4 Animation

\(\text{Normal density function}\)

\(\mathbf{ \color{green}{r(x) = e^{\frac{-(x-\color{red}{A} \color{green}{ )^2} }{b} } } }\)
gallery_normaldensity_pic Vary A MP4 Animation

Vary b MP4 Animation


Selected Resources

  1. Transcendental Function
  2. Some History
  3. Exp Function
  4. Exponentiation
  5. more Exp Function
  6. Math Constant
  7. How to use a slide rule
  8. Early calculator
  9. History of logarithms
  10. Britannica logarithm
  11. Further details
  12. Logistic equation
  13. Normal distribution
  14. Where does e come from?
  15. Exp Function
  16. Exp Function again
  17. History Computer
  18. Slide Rule