Question(1 point)
Solve for :
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Studdy Solution
STEP 1
1. The equation involves logarithms with the same base, which allows us to use properties of logarithms to simplify and solve.
2. The domain of the logarithmic function requires that the arguments be positive, i.e., and .
STEP 2
1. Simplify the equation using properties of logarithms.
2. Solve the resulting equation for .
3. Verify the solution(s) against the domain restrictions.
STEP 3
Start by simplifying the equation using properties of logarithms. Notice that simplifies to :
STEP 4
Use the property of logarithms that allows us to express as :
STEP 5
To eliminate the constant term, rewrite the equation as:
Convert the constant 2 to a logarithmic form: , because :
STEP 6
Apply the property of logarithms that allows the addition of logs to be expressed as a single log of a product:
Since the logarithms are equal, their arguments must be equal:
STEP 7
Solve the quadratic equation :
First, expand and rearrange the equation:
STEP 8
Factor the quadratic equation:
Set each factor equal to zero and solve for :
STEP 9
Verify the solutions against the domain restrictions:
- For , .
- For , .
Both and satisfy these conditions.
The solutions for are:
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