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Does a linear, exponential, or logarithmic model best fit the data in [link] ? Find the model.

x 1 2 3 4 5 6 7 8 9
y 3.297 5.437 8.963 14.778 24.365 40.172 66.231 109.196 180.034

Exponential. y = 2 e 0.5 x .

Expressing an exponential model in base e

While powers and logarithms of any base can be used in modeling, the two most common bases are 10 and e . In science and mathematics, the base e is often preferred. We can use laws of exponents and laws of logarithms to change any base to base e .

Given a model with the form y = a b x , change it to the form y = A 0 e k x .

  1. Rewrite y = a b x as y = a e ln ( b x ) .
  2. Use the power rule of logarithms to rewrite y as y = a e x ln ( b ) = a e ln ( b ) x .
  3. Note that a = A 0 and k = ln ( b ) in the equation y = A 0 e k x .

Changing to base e

Change the function y = 2.5 ( 3.1 ) x so that this same function is written in the form y = A 0 e k x .

The formula is derived as follows

y = 2.5 ( 3.1 ) x = 2.5 e ln ( 3.1 x ) Insert exponential and its inverse . = 2.5 e x ln 3.1 Laws of logs . = 2.5 e ( ln 3.1 ) x Commutative law of multiplication

Change the function y = 3 ( 0.5 ) x to one having e as the base.

y = 3 e ( ln 0.5 ) x

Key equations

Half-life formula If   A = A 0 e k t , k < 0 , the half-life is   t = ln ( 2 ) k .
Carbon-14 dating t = ln ( A A 0 ) 0.000121 .
A 0   A   is the amount of carbon-14 when the plant or animal died
t   is the amount of carbon-14 remaining today
is the age of the fossil in years
Doubling time formula If   A = A 0 e k t , k > 0 , the doubling time is   t = ln 2 k
Newton’s Law of Cooling T ( t ) = A e k t + T s , where   T s   is the ambient temperature,   A = T ( 0 ) T s , and   k   is the continuous rate of cooling.

Key concepts

  • The basic exponential function is f ( x ) = a b x . If b > 1 , we have exponential growth; if 0 < b < 1 , we have exponential decay.
  • We can also write this formula in terms of continuous growth as A = A 0 e k x , where A 0 is the starting value. If A 0 is positive, then we have exponential growth when k > 0 and exponential decay when k < 0. See [link] .
  • In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay. See [link] .
  • We can find the age, t , of an organic artifact by measuring the amount, k , of carbon-14 remaining in the artifact and using the formula t = ln ( k ) 0.000121 to solve for t . See [link] .
  • Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay. See [link] .
  • We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time. See [link] .
  • We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors. See [link] .
  • We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data. See [link] .
  • Any exponential function with the form y = a b x can be rewritten as an equivalent exponential function with the form y = A 0 e k x where k = ln b . See [link] .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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.
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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