PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

Example 4: Solve for x in the equation

Solution:

Step 1: Note that for the problem to be valid, we should have

Therefore, the problem is valid if we restrict the domain (values of x) to any real numbers x < - 2 or x > - 1. You can also graph the function

and note that there is a graph to the right of - 1 and to the left of - 2; there is no graph between these values. Note that the graph has two intercepts, one positive and one negative. The graph crosses the x-axis between 8 an 9, so you know one answer will be between 8 and 9. The graph also crosses the x-axis between - 11 and - 12, so you one answer will be between these two numbers.

Step 2: Rewrite the equation as

Step 3: Simplify the left side of the above equation:

Step 4: We need to isolate the log term, so multiply both sides by 2:

Step 5: Convert the above equation to an exponential equation:

Step 6: Subtract 100 from both sides of the above equation:

Step 7: Solve for x with the quadratic formula

which gives

Check: You can check your answer in two ways. You could graph the function

and see where it crosses the x-axis. If you are correct, the graph should cross the x-axis at 8.512492197 and - 11.15124921973.
You can also check your answer by substituting the values of x in the initial equation and determining whether the left side of the equation equals the right side of the equation after the substitution.

If you would like to review another example, click on Example.

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