Question
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Answer
The solution set is {9}.
Solution
We start with the given equation:
Step 1: Combine the logarithms
Using the logarithm property
we rewrite the equation as:
Step 2: Equate the arguments
Since the logarithm function is one-to-one, if
then
So we have:
Step 3: Form the quadratic equation
Expanding the left side:
Subtract 27 from both sides to set the equation to zero:
Step 4: Solve the quadratic equation
We calculate the discriminant:
Since
, the solutions are given by:
Thus, we find two potential solutions:
Step 5: Check the domain restrictions
For the original logarithmic expressions, the arguments must be positive:
- For
, we require . - For
, we require or .
The value
does not satisfy
and
.
Thus, the only valid solution is
.
Conclusion:
The solution set is
.
The correct choice is A.
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Mind Expander
To solve the equation
, we can use the properties of logarithms. By applying the product property, we can rewrite the left side:
From this, we equate the arguments since the logarithmic function is one-to-one:
Expanding this gives:
Now we can use the quadratic formula to find the roots:
This gives us two potential solutions:
Since logarithms are only defined for positive numbers, we discard
. Thus, the only solution is
.
So the solution set is:
A. The solution set is {9}.