# 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.