11.4 Notes Solving logarithmic equations

11.4 Notes In this unit of study, you will learn several methods for solving several types of logarithmic equations. In a previous lesson, you learned how to solve logarithmic equations that have only one logarithm term. In this lesson, you will learn how to solve logarithmic equations that have more than one logarithm term.

11.4 Notes The objective when solving logarithmic equations with more than one term in them is to try to get one logarithm term, with matching bases, on each side of the equal sign: Once this has been achieved, the logarithm terms can be ignored, the arguments of the terms set equal to each other and solved:

11.4 Notes Before answers to logarithmic equations can be declared the solutions, it must be determined whether they satisfy the original equation. Because the range of is, the domain of is. Therefore, when answers to logarithmic equations are substituted back into the original equation, the arguments of the equation must be in the interval

11.4 Notes When -2 is substituted back into the original equation, the argument of the logarithm terms on each side of the equal sign is +4; therefore, -2 is the solution to the equation.

11.4 Notes – Example 1 When -3 is substituted back into the original equation, the argument of the logarithm term on the left side of the equal sign is -28; therefore, -3 is not a solution. This equation has no solution.

11.4 Notes – Practice 1 When 2 is substituted back into the original equation, the argument of the logarithm term on the left side of the equal sign is -7; therefore, 2 is not a solution. This equation has no solution.

11.4 Notes The objective when solving logarithmic equations with more than one term in them is to try to get one logarithm term, with matching bases, on each side of the equal sign. When a logarithmic equation contains more than one logarithm term on either side of the equation, it is necessary to get only one logarithm term, with matching bases, on each side of the equal sign. Doing so requires the use of properties of logarithms.

11.4 Notes Product property: Quotient property: Power property:

11.4 Notes – Example 2

11.4 Notes – Practice 2

11.4 Notes – Example 3

11.4 Notes – Example 4

11.4 Notes – Practice 3

11.4 Notes – Example 5

11.4 Notes – Practice 4