# Partial Fraction Decomposition

Example:  Find the partial fraction decomposition for the rational expression.

Solution:

 The rational expression Factor the denominator completely Write the expression with each factor as a separate fraction. Decide what type of expression to put in the numerator x is linear so we put a constant in the numerator is a quadratic so we put a linear expression in the numerator Now that the set up is done we need to solve for the unknowns A, B and C Multiply both sides of the equation by the common denominator and simplify by canceling out common factors Use the distributive property and collect the like terms with the x’s Equate coefficients to create a system of equations to solve On the left 11 is the constant term.  There is no linear term so the coefficient is zero and there is no quadratic term so the coefficient is zero. On the right the constant (no x’s) is 11A, the linear coefficient is C and the quadratic coefficient is A+B. System of equations from equating the coefficients

Now solve the system of equations.

The first equation is  11=11A.  Since it only has one variable, I can solve for A by hand.

The second equation is 0=C .  It is already solved.

The third equation is 0=A+B.  Since I know A=1, I can substitute and solve for B.

Using the original set up of , substitute the values of A, B, and C.

The partial fraction decomposition of is

# 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.