If we are given an equation with a logarithm of the same base on both sides we may simply equate the arguments.
Step 1: Use the rules of exponents to isolate a logarithmic expression (with the same base) on both sides of the equation. Step 2: Set the arguments equal to each other. Step 3: Solve the resulting equation. Step 4: Check your answers.
Be sure to check to see if the solutions that you obtain solve the original logarithmic equation. In this study guide we will put a check mark next to the solution after we determine that it really does solve the equation. This process sometimes results in extraneous solutions so we must check our answers. Solve.
Of course, equations like these are very special. Most of the problems that we will encounter will not have 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 2: 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 extraneous. This does not preclude negative answers. We must be sure to check all of our solutions.
Tip: Not all negative solutions are extraneous! Look at the previous set of problems and see that some have negative answers. The check mark indicates that we actually plugged the answers in to see that they do indeed solve the original. Please do not skip this step, extraneous solutions occur often.
Since multiplication is commutative, you can multiply the coefficients and the radicands together and then simplify.
Multiply.
Take care to be sure that the indices are the same before multiplying. We will assume that all variables are positive.
Simplify.
Divide radicals using the following property.
Divide. (Assume all variables are positive.)
Rationalizing the Denominator
A simplified radical expression cannot have a radical in the denominator. When the denominator has a radical in it, we must multiply the entire expression by some form of 1 to eliminate it. The basic steps follow.
Rationalize the denominator:
Multiply numerator and denominator by the 5th root of of factors that will result in 5th powers of each factor in the radicand of the denominator.
Notice that all the factors in the radicand of the denominator have powers that match the index. This was the desired result, now simplify.
Rationalize the denominator.
This technique does not work when dividing by a binomial containing a radical. A new technique is introduced to deal with this situation.
Rationalize the denominator:
Multiply numerator and denominator by the conjugate of the denominator.
And then simplify. The goal is to eliminate all radicals from the denominator.
