# Application: Exponential Growth and Decay (Half-life)

Example: The half life of radium is 1690 years. If 50 grams are present now, how much will be present in 630 years?

Solution:  There is a two part process to this problem.  Part 1: Use some of the information to find the decay rate of radium.  Part 2: Answer the question using the rest of the given information.

Part 1:  Find the decay rate of radium.

Since we are using an exponential model for this problem we should be clear on the parts of the exponential decay model.

Exponential Decay Model

is the initial amount

is the decay rate

is the time

is the amount after t time has passed

Since radium has a half life of 1690 years, we know that whatever initial amount of radium is present after 1690 years there will be half of that initial amount left.  This allows me to identify when the .

Substitute these values into the exponential decay formula and solve for k.

 Exponential Decay Formula Make a substitution for A and t since it is known that the half-life is 1690 years and Solve for the decay rate k:Start by dividing both sides by the coefficient to isolate the exponential factor Solve for the decay rate k:Take the natural log of both sides to get k out of the exponent Solve for the decay rate k:Use the power rule for logarithms to get k out of the exponent Solve for the decay rate k:Simplify ln e = 1 Solve for the decay rate k:Solve for k by dividing by 1690 on both sides

Using only the information about radium having a half-life of 1690 years I have found the decay rate for radium.

Note: Although I have put an approximation for k here, try not to round until the very last step.

Part 2:  Answer the question using the rest of the given information.

Given information: If 50 grams are present now, how much will be present in 630 years?

With this information I can identify the initial amount of radium as 50 grams and the time to be 630 years.  Symbolically that is when the .  Substitute the given information and the decay rate k found in part 1 to the exponential decay formula.

 Exponential Decay Formula Make a substitution for initial amount , time (t), and the decay rate (k). , , and Type the expression into your calculator and round to the thousandth place.

38.615 grams will be present 630 years later is 50 grams are present initially.

# Application of Exponential Functions: Finding the Interest Rate

Example:

What is the interest rate necessary for an investment to quadruple after 7 year of continuous compound interest?

Solution:

Since this question involve continuous compound interest, we will use the associated formula.

We are given that the invest quadruples in 7 years.  This tells me that when t=7 that A will be 4 times P.  I can write that in symbols A=4P.

Substitute these values into the continuous compound formula and solve for the interest rate.

 Continuous compound formula Substitute the values of t and A into the formula Solve for r by dividing both sides by P and simplifying Solve for r by taking the log of both sides. Solve for r by using the power rule and simplifying Solve for r by dividing both sides by 7 and simplifying Find the value in the calculator Write the answer as a percentage rounded to two decimal places

# Solve an exponential equation: Take the log of both sides

Example:

Solution:

 The exponential equation Since the bases cannot be easily written the same use the method of taking the log of both sides Use the power rule for logarithms Use the distributive law Collect the terms with x to one side and collect the terms without x on the other side Factor the common x Solve for x by dividing both sides by the factor in the parenthesis and simplify The solution

When you type this into a calculator be sure to use parenthesis around the numerator and around the denominator.  Here is an example of how you might enter it.

(ln (sqrt{2}))/(ln(1.41)+4 ln(sqrt{2}))

# Solve the Logarithmic Equation by the one to one property

Example:

Solution:

 The logarithmic equation Use the power rule and the quotient rule to condense to a single logarithm Since both sides of the equation have the same log base the expressions inside the logarithms must be equal Clear the denominator by multiplying by 2 on both sides and simplifying Get rid of the square by square rooting both sides and simplifying Get x by itself by subtracting 7 on both sides Get x by itself by dividing both sides by negative 1 or or Check x=13 Log of a negative is undefined.  Exclude this solution. Check x=1 Keep this solution.

The solution to the equation is x=1.

# Solve an Exponential Equation: Take the log of both sides

Problem:  Solve the exponential equation.

Solution:

 The exponential equation Since the bases cannot be easily written the same, use the method of taking the log of both sides Use the power rule for logarithms. Use the distributive law Collect the terms with x to one side and collect the terms without x on the other side Factor the common x Solve for x by dividing both sides by the factor in the parenthesis and simplify The solution

When you type this into a calculator be sure to use parenthesis around the numerator and around the denominator.  Here is an example of how you might enter it.

(3ln(16)-3ln(3))/(3ln(16)-ln(3))

# Solving an Exponential Equation: Relating the Bases

Example:  Solve the exponential equation.

Solution:

 The exponential equation Try to write both sides of the equation with the same base.  Try 4 since there is a base of 4 on the left Using a property of negative exponents move the base to the numerator Now that that the bases are the same the exponents must be equal Solve for x

The solution the the exponential equation is 4.

# Graphing an Exponential Equation by Transformations

Example: For the function below.  Graph using transformations.  Find the y-intercept.  State the horizontal asymptote and the domain and range.

First we must examine the base function

Graph using plotting points.  We can use the standard set of x-values to find ordered pairs.

xy
-22^(-2)=1/4
-12^(-1)=1/2
02^0=1
12^1=2
22^2=4

The graph below shows the points plotted and the line that connects them.  This graph has a horizontal asymptote at y=0.  The domain is   and the range is

Analyze the transformations.

The +2 in the exponent shifts the graph left 2 units.

The – in the front of the base reflects the graph over the x-axis.

The +2 next to the base shifts the graph and the horizontal asymptote up two units.

You can see the graph after the transformations.

The horizontal asymptote is y=2.  The domain is   and the range is

To find the y-intercept we let x=0.

Thus the y-intercept is (0,-2)