PROPERTIES OF LOGARITHMS
SOLVING LOGARITHMIC EQUATIONS
1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.
Problem 4: Solve for x in the equation
Answer: x = 2
- As you know by now, we can only take the logarithm of a positive number. Therefore, we will have to restrict the domain (values of x) so that the original equation is valid.
- Step 1: The term is valid when x is greater than zero, and the term is valid when x is greater than 1. If we restrict the domain to the set of all real numbers that are greater than 1, both terms will be valid.
- Step 2: Simplify the left side of the original equation using Logarithmic Rule 2:
- Step 3: Convert the logarithmic term to an exponential term with base 4 and exponent :
- Step 4: Simplify the left side of the above equation:
- Step 5: Multiply both sides of the above equation by (x - 1):
- Step 6: Expand the left side of the above equation:
- Step 7: Subtract x from both sides of the above equation and add 2 to both sides of the above equation:
Check: Let's substitute the value x = 2 in the original equation and
determine whether the left side of that equation equals the right side of
that equation after the substitution
Since the value of the left side of the equation is
after you substitute the value 2 for x, and the right side of the equal is , you have proved your answer.
If you would like to work on another problem, click on
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