## SOLVING LOGARITHMIC EQUATIONS

Note:

If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic functions.

Solve for x in the following equation.

Problem 8.4d:

and The approximate answers are 1.2478358458, 4.75216415412, and

Solution:

The above equation is valid only if the expression The expression is valid when and Another way of saying this is that the domain is the set of real numbers where

Note: If, when solving the above problem, you simplify the logarithmic equation to you will lose half of your solutions. Why?

Recall that is valid for all real numbers that are not equal to 1 or 5. However, is valid for the set of real numbers This means that only over the set of real numbers

So be careful when you simplify a logarithmic equation before solving it.

Convert the logarithmic equation to an equivalent exponential equation with base 0.982.

These answers may or may not be the solutions. You must check them numerically or graphically.

Numerical Check:

• Check the answer by substituting 5.22034249092 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 5.22034249092 for x, then is a solution.

• Check the answer by substituting 4.75216415413 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 4.75216415413 for x, then is a solution.

• Check the answer by substituting 0.779657509082 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 0.779657509082 for x, then is a solution.

• Check the answer by substituting 1.24783584587 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Left Side:

Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 1.24783584587 for x, then is a solution.

All four of the answers are the solutions to the original equation.

Graphical Check:

You can also check your answer by graphing (formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; that will be the real solution. Note that the graph crosses the x-axis at four places: , and 5.22034249092. This means that , and 5.22034249092 are the real solutions.

If you have trouble graphing the function , graph the equivalent function .

If you would like to go to the next section, click on next.

If you would like to go back to the previous section, click on previous.

This site was built to accommodate the needs of students. The topics and problems are what students ask for. We ask students to help in the editing so that future viewers will access a cleaner site. If you feel that some of the material in this section is ambiguous or needs more clarification, or you find a mistake, please let us know by e-mail.

[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Nancy Marcus