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# Exponential Models

## Population and investing using exponential functions.

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Exponential and Logarithmic Models

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### Vocabulary

##### Complete the table.
 Word Definition ______________ to create new data points, or to predict, outside of the domain of the data set Interpolate ______________________________________________________________________ ______________ a solution to a function that is not useful in the given context Exponential Model ______________________________________________________________________

### Models

A population which increases continuously at a constant rate may be modeled with an exponential function.

A population which increases rapidly and then levels off may be modeled with a logarithmic function.

#### Exponential

The equation for population growth is: P(f)=Pirx\begin{align*}P(f) = P_i \cdot r^{x}\end{align*}

What does each letter in the equation mean? ______________  ______________  ______________  ______________

This equation is used in countless situations besides population growth. What are some situations in which you would use that equation? ________________________________________________________

.

For problems 1-3, calculate:

a) The growth factor

b) The final population

1. If a population starts at 5,000 people in 1990, and increases at a rate of 9% per year, what is the population in 2037?
2. If a population starts at 15,000 people in 2000, and increases at a rate of 6% per year, what is the population in 2019?
3. If a population starts at 25,500 people in 1900, and increases at a rate of 4% per year, what is the population in 2004?

#### Logarithmic

To solve complex logarithmic equations, you must use your knowledge from algebra as well as the logarithmic properties

What are three logarithmic properties?

1) ____________________________________

2) ____________________________________

3) ____________________________________

.

Solve for x\begin{align*}x\end{align*}

1. 4log(x5)+log(6254)=2logx\begin{align*}4 log (\frac{x}{5}) + log (\frac{625}{4}) = 2 log x\end{align*}
2. log5z+log5125log5x=72\begin{align*}log_5 z + \frac{log_5 125}{log_5 x} = \frac{7}{2}\end{align*}
3. logp=2logplogp\begin{align*}log p = \frac{2 - log p}{log p}\end{align*}
4. 2logx2log(x+1)=0\begin{align*}2 log x - 2 log (x+1) = 0\end{align*}
5. log(25z3)3log(4z)=0\begin{align*}log (25 - z^3) - 3log (4 - z) = 0\end{align*}
6. log(35y3)log(5y)=3\begin{align*}\frac{log (35 - y^3)}{log (5 - y)} = 3\end{align*}

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