# Precalculus : Exponential Functions

## Example Questions

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### Example Question #1 : Exponential Functions

Suppose the graph of an exponential equation contains the points  and . What is the formula of this line?

Explanation:

Recall the standard form of an exponential equation:

Plug in the two ordered pairs:

Solve for a in the first equation:

Now plug this value into the second equation:

Simplify:

Multiply both sides by 3/2 and simplify:

Plug this into the first equation to solve for a:

Plug the values of a and b into the general form to get the answer:

### Example Question #1 : Exponential Functions

A calculator is required to solve this problem.

Suppose a population is currently at 5000, and that it increases by 5% every year. How large will the population be in 5 years? Round your answer to the nearest whole number.

6381

5000

5469

6250

5250

6381

Explanation:

The standard formula for population growth is

,

where  is the population after time t,  is the initial population, and r is the rate of growth as a decimal, per unit time. (In other words, r and t must have the same units.)

The problem also gives us initial population as 5000 and growth rate as 5% or .05.

Plug in:

Simplify the parentheses:

Evaluate:

Round to get the final answer:

### Example Question #1 : Exponential Functions

To solve this question you will need a calculator or other graphing tool capable of evaluating logarithms.

Suppose a colony of bacteria is decaying at a constant rate of 2% per minute. How many minutes will it take for the colony's population to decrease by half? Round your answer to the nearest whole minute.

25 minutes

20 minutes

-35 minutes

34 minutes

35 minutes

34 minutes

Explanation:

We recall that the formula for population decay is

,

where  is the population at time t,  is the initial population, and r is the rate of decrease per unit time (same unit as t).

is half of , so we can write

.

Simplify and eliminate common factors:

Take the log of both sides. Note that the log has base .98.

Use the change of base theorem to rewrite the log:

Round:

### Example Question #4 : Exponential Functions

Expand and simplify:

Explanation:

### Example Question #1 : Exponential Functions

Identify the curve representing  in the graph below.

B

E

C

A

D

D

Explanation:

the -int will be when  and the -int will not exist because all the values of this function are positive.

So the curve must be D since that is the only curve that intersects at the point

### Example Question #1 : Solve Exponential Equations

Solve an equation involving exponents and logarithms.

Solve for .

Explanation:

First, simplify the left side of the equation using the additive rule for exponents.

.

Our equation now becomes:

Equating we set the exponents equal to eachother and solve.

Thus,

### Example Question #121 : Exponential And Logarithmic Functions

Solve an exponential equation.

Solve for .

Explanation:

First, use the additive property of exponents to simplify the right side of the equation.

.

Thus,

.

Now, take the natural log of both sides

.

Use the multiplicative property of logarithms to expand the left side to get

Now, apply the logarithms to the exponents

.

Rearrange to get the x-terms on one side

.

Finally, divide the 2 on both sides

.

### Example Question #2 : Solve Exponential Equations

Solve for

Explanation:

First, let's begin by simplifying the left hand side.

becomes   and  becomes . Remember that , and the  in that expression can come out to the front, as in .

Now, our expression is

From this, we can cancel out the 2's and an x from both sides.

.

### Example Question #4 : Solve Exponential Equations

The population of fish in a pond is modeled by the exponential function

, where  is the population of fish and  is the number of years since January 2010.

Determine the population of fish in January 2010 and January 2015.

2010:  fish

2015:  fish

2010:  fish

2015:  fish

2010:  fish

2015:  fish

2010:  fish

2015:  fish

2010:  fish

2015:  fish

Explanation:

In 2010,  in our equation because we have had no years past 2010. Plugging that in to the model equation and solving:

, since anything raised to the power of zero becomes . So the population of fish in 2010 is  fish.

In 2015,  because 5 years have passed since 2010. Plugging that into our equation and solving gives us

So the population of fish in 2015 is  fish. This is an example of exponential decay since the function is decreasing.

### Example Question #1 : Solve Exponential Equations

Solve for  using properties of exponents.