While powers and logarithms of any base can be used in modeling, the two most common bases are
and
In science and mathematics, the base
is often preferred. We can use laws of exponents and laws of logarithms to change any base to base
Given a model with the form
change it to the form
Rewrite
as
Use the power rule of logarithms to rewrite y as
Note that
and
in the equation
Changing to base
e
Change the function
so that this same function is written in the form
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
the doubling time is
Newton’s Law of Cooling
where
is the ambient temperature,
and
is the continuous rate of cooling.
Key concepts
The basic exponential function is
If
we have exponential growth; if
we have exponential decay.
We can also write this formula in terms of continuous growth as
where
is the starting value. If
is positive, then we have exponential growth when
and exponential decay when
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,
of an organic artifact by measuring the amount,
of carbon-14 remaining in the artifact and using the formula
to solve for
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
can be rewritten as an equivalent exponential function with the form
where
See
[link] .
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
