While powers and logarithms of any base can be used in modeling, the two most common bases are
$\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ In science and mathematics, the base
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ is often preferred. We can use laws of exponents and laws of logarithms to change any base to base
$\text{\hspace{0.17em}}e.$
Given a model with the form
$\text{\hspace{0.17em}}y=a{b}^{x},$ change it to the form
$\text{\hspace{0.17em}}y={A}_{0}{e}^{kx}.$
Rewrite
$\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ as
$\text{\hspace{0.17em}}y=a{e}^{\mathrm{ln}\left({b}^{x}\right)}.$
Use the power rule of logarithms to rewrite y as
$\text{\hspace{0.17em}}y=a{e}^{x\mathrm{ln}\left(b\right)}=a{e}^{\mathrm{ln}\left(b\right)x}.$
Note that
$\text{\hspace{0.17em}}a={A}_{0}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}k=\mathrm{ln}\left(b\right)\text{\hspace{0.17em}}$ in the equation
$\text{\hspace{0.17em}}y={A}_{0}{e}^{kx}.$
Changing to base
e
Change the function
$\text{\hspace{0.17em}}y=2.5{(3.1)}^{x}\text{\hspace{0.17em}}$ so that this same function is written in the form
$\text{\hspace{0.17em}}y={A}_{0}{e}^{kx}.$
Change the function
$\text{\hspace{0.17em}}y=3{(0.5)}^{x}\text{\hspace{0.17em}}$ to one having
$\text{\hspace{0.17em}}e\text{\hspace{0.17em}}$ as the base.
$y=3{e}^{\left(\mathrm{ln}0.5\right)x}$
Access these online resources for additional instruction and practice with exponential and logarithmic models.
If
$\text{}A={A}_{0}{e}^{kt},$$k<0,$ the half-life is
$\text{}t=-\frac{\mathrm{ln}(2)}{k}.$
Carbon-14 dating
$t=\frac{\mathrm{ln}\left(\frac{A}{{A}_{0}}\right)}{-0.000121}.$ ${A}_{0}\text{}$$A\text{}$ is the amount of carbon-14 when the plant or animal died
$t\text{}$ is the amount of carbon-14 remaining today
is the age of the fossil in years
Doubling time formula
If
$\text{}A={A}_{0}{e}^{kt},$$k>0,$ the doubling time is
$\text{}t=\frac{\mathrm{ln}2}{k}$
Newton’s Law of Cooling
$T(t)=A{e}^{kt}+{T}_{s},$ where
$\text{}{T}_{s}\text{}$ is the ambient temperature,
$\text{}A=T(0)-{T}_{s},$ and
$\text{}k\text{}$ is the continuous rate of cooling.
Key concepts
The basic exponential function is
$\text{\hspace{0.17em}}f(x)=a{b}^{x}.\text{\hspace{0.17em}}$ If
$\text{\hspace{0.17em}}b>1,$ we have exponential growth; if
$\text{\hspace{0.17em}}0<b<1,$ we have exponential decay.
We can also write this formula in terms of continuous growth as
$\text{\hspace{0.17em}}A={A}_{0}{e}^{kx},$ where
$\text{\hspace{0.17em}}{A}_{0}\text{\hspace{0.17em}}$ is the starting value. If
$\text{\hspace{0.17em}}{A}_{0}\text{\hspace{0.17em}}$ is positive, then we have exponential growth when
$\text{\hspace{0.17em}}k>0\text{\hspace{0.17em}}$ and exponential decay when
$\text{\hspace{0.17em}}k<0.\text{\hspace{0.17em}}$ 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,
$\text{\hspace{0.17em}}t,$ of an organic artifact by measuring the amount,
$\text{\hspace{0.17em}}k,$ of carbon-14 remaining in the artifact and using the formula
$\text{\hspace{0.17em}}t=\frac{\mathrm{ln}\left(k\right)}{-0.000121}\text{\hspace{0.17em}}$ to solve for
$\text{\hspace{0.17em}}t.\text{\hspace{0.17em}}$ 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
$\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ can be rewritten as an equivalent exponential function with the form
$\text{\hspace{0.17em}}y={A}_{0}{e}^{kx}\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}k=\mathrm{ln}b.\text{\hspace{0.17em}}$ See
[link] .
Questions & Answers
where we get a research paper on Nano chemistry....?
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?