# 4.7 Exponential and logarithmic models  (Page 7/16)

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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. $\text{\hspace{0.17em}}y=2{e}^{0.5x}.$

## 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 $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}y={A}_{0}{e}^{kx}.$

1. Rewrite $\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}y=a{e}^{\mathrm{ln}\left({b}^{x}\right)}.$
2. Use the power rule of logarithms to rewrite y as $\text{\hspace{0.17em}}y=a{e}^{x\mathrm{ln}\left(b\right)}=a{e}^{\mathrm{ln}\left(b\right)x}.$
3. 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 $\text{\hspace{0.17em}}y=2.5{\left(3.1\right)}^{x}\text{\hspace{0.17em}}$ so that this same function is written in the form $\text{\hspace{0.17em}}y={A}_{0}{e}^{kx}.$

The formula is derived as follows

Change the function $\text{\hspace{0.17em}}y=3{\left(0.5\right)}^{x}\text{\hspace{0.17em}}$ 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.

## Key equations

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

## Key concepts

• The basic exponential function is $\text{\hspace{0.17em}}f\left(x\right)=a{b}^{x}.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}b>1,$ we have exponential growth; if $\text{\hspace{0.17em}}0 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] .

how fast can i understand functions without much difficulty
what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this