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exp ( ln x ) = x for x > 0 and ln ( exp x ) = x for all x .

The following figure shows the graphs of exp x and ln x .

This figure is a graph. It has three curves. The first curve is labeled exp x. It is an increasing curve with the x-axis as a horizontal asymptote. It intersects the y-axis at y=1. The second curve is a diagonal line through the origin. The third curve is labeled lnx. It is an increasing curve with the y-axis as an vertical axis. It intersects the x-axis at x=1.
The graphs of ln x and exp x .

We hypothesize that exp x = e x . For rational values of x , this is easy to show. If x is rational, then we have ln ( e x ) = x ln e = x . Thus, when x is rational, e x = exp x . For irrational values of x , we simply define e x as the inverse function of ln x .


For any real number x , define y = e x to be the number for which

ln y = ln ( e x ) = x .

Then we have e x = exp ( x ) for all x , and thus

e ln x = x for x > 0 and ln ( e x ) = x

for all x .

Properties of the exponential function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of e , we must verify that the usual laws of exponents hold for the function e x .

Properties of the exponential function

If p and q are any real numbers and r is a rational number, then

  1. e p e q = e p + q
  2. e p e q = e p q
  3. ( e p ) r = e p r


Note that if p and q are rational, the properties hold. However, if p or q are irrational, we must apply the inverse function definition of e x and verify the properties. Only the first property is verified here; the other two are left to you. We have

ln ( e p e q ) = ln ( e p ) + ln ( e q ) = p + q = ln ( e p + q ) .

Since ln x is one-to-one, then

e p e q = e p + q .

As with part iv. of the logarithm properties, we can extend property iii. to irrational values of r , and we do so by the end of the section.

We also want to verify the differentiation formula for the function y = e x . To do this, we need to use implicit differentiation. Let y = e x . Then

ln y = x d d x ln y = d d x x 1 y d y d x = 1 d y d x = y .

Thus, we see

d d x e x = e x

as desired, which leads immediately to the integration formula

e x d x = e x + C .

We apply these formulas in the following examples.

Using properties of exponential functions

Evaluate the following derivatives:

  1. d d t e 3 t e t 2
  2. d d x e 3 x 2

We apply the chain rule as necessary.

  1. d d t e 3 t e t 2 = d d t e 3 t + t 2 = e 3 t + t 2 ( 3 + 2 t )
  2. d d x e 3 x 2 = e 3 x 2 6 x
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Evaluate the following derivatives:

  1. d d x ( e x 2 e 5 x )
  2. d d t ( e 2 t ) 3
  1. d d x ( e x 2 e 5 x ) = e x 2 5 x ( 2 x 5 )
  2. d d t ( e 2 t ) 3 = 6 e 6 t
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Using properties of exponential functions

Evaluate the following integral: 2 x e x 2 d x .

Using u -substitution, let u = x 2 . Then d u = −2 x d x , and we have

2 x e x 2 d x = e u d u = e u + C = e x 2 + C .
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Evaluate the following integral: 4 e 3 x d x .

4 e 3 x d x = 4 3 e −3 x + C

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General logarithmic and exponential functions

We close this section by looking at exponential functions and logarithms with bases other than e . Exponential functions are functions of the form f ( x ) = a x . Note that unless a = e , we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function f ( x ) = a x in terms of the exponential function e x . We then examine logarithms with bases other than e as inverse functions of exponential functions.


For any a > 0 , and for any real number x , define y = a x as follows:

y = a x = e x ln a .

Now a x is defined rigorously for all values of x . This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of r . It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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