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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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
how nano science is used for hydrophobicity
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
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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