Solving Logarithmic Equations Extraneous Solutions

Use the one-to-i property for logarithms to solve logarithmic equations.

If nosotros are given an equation with a logarithm of the same base on both sides we may merely equate the arguments.

Stride i: Use the rules of exponents to isolate a logarithmic expression (with the same base of operations) on both sides of the equation.
Step two: Set up the arguments equal to each other.
Stride 3: Solve the resulting equation.
Step iv: Bank check your answers.

Be sure to bank check to see if the solutions that you lot obtain solve the original logarithmic equation. In this study guide we volition put a check marker next to the solution after we make up one's mind that it really does solve the equation. This process sometimes results in extraneous solutions and then we must check our answers.
Solve.

Of course, equations similar these are very special. Most of the problems that nosotros will encounter will not accept a logarithm on both sides. The steps for solving them follow.

Step 1: Use the properties of the logarithm to isolate the log on one side.
Step ii: Apply the definition of the logarithm and rewrite it as an exponential equation.
Step 3: Solve the resulting equation.
Step 4: Check your answers.

If the answer to the logarithmic equation makes the argument negative then it is inapplicable. This does not preclude negative answers. We must exist sure to check all of our solutions.

Instructional Video: Solving Logarithmic Equations

Solve.

Tip: Non all negative solutions are extraneous! Look at the previous set of bug and see that some have negative answers. The bank check mark indicates that we really plugged the answers in to see that they exercise indeed solve the original. Please do not skip this stride, extraneous solutions occur often.