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
- 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
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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