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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?