PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

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

Example 7: Solve for x in terms of b in the equation

Solution:

Step 1: The term is valid when , and the term is valid when x>0. If we restrict the domain to the set of all real numbers x between 0 and , every term in the equation is valid.
You can also graph the function

using a positive value for b and note that the entire graph is located between the values x = 0 and .
Note also that the value of the base b must be greater than zero.

Step 2: Simplify the original equation by gathering the logarithmic terms to the left side of the equal sign:

Step 3: Simplify the left side of the equation using Logarithmic Rule 2:

Step 4: Convert the above equation to an exponential equation with base b and exponent 3:

Step 5: Multiply both sides of the above equation by x:

Step 6: Add 3x to both sides of the above equation:

Step 7: Factor the left side of the above equation:

Step 8: Divide both sides of the above equation by :

You have solved for x in terms of b.

Check: Let's check the answer by substituting

in the original equation. If, after the substitution, the left side of the original equations equals the right side of the original equation, you have found the right answer: Does

The term

can be simplified to

which is equal to the right side of the equation..
You have correctly worked the problem.

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

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