To solve the equation [tex]\( \log_3(15 - p) = \log_3(p + 3) \)[/tex], we can use the properties of logarithms and step-by-step algebraic manipulation. Here's a detailed solution:
1. Understand the equation: The given equation is
[tex]\[
\log_3(15 - p) = \log_3(p + 3)
\][/tex]
2. Use the properties of logarithms: When the logarithm of a quantity to the same base is equal, the quantities inside the logarithms must also be equal. Therefore, we can set the arguments equal to each other:
[tex]\[
15 - p = p + 3
\][/tex]
3. Solve for [tex]\( p \)[/tex]:
- Start by isolating [tex]\( p \)[/tex]. Subtract [tex]\( p \)[/tex] from both sides of the equation:
[tex]\[
15 - p - p = 3
\][/tex]
- Simplify the equation:
[tex]\[
15 - 2p = 3
\][/tex]
- Isolate [tex]\( p \)[/tex] by first subtracting 15 from both sides:
[tex]\[
-2p = 3 - 15
\][/tex]
- Simplify:
[tex]\[
-2p = -12
\][/tex]
- Finally, divide by -2 to solve for [tex]\( p \)[/tex]:
[tex]\[
p = 6
\][/tex]
4. Verify the solution:
- Check the initial constraints for the problem: The arguments [tex]\( 15 - p \)[/tex] and [tex]\( p + 3 \)[/tex] must be positive for the logarithms to be defined.
- For [tex]\( p = 6 \)[/tex]:
- [tex]\( 15 - 6 = 9 \)[/tex] which is positive.
- [tex]\( 6 + 3 = 9 \)[/tex] which is positive.
- Both of these conditions are satisfied, so [tex]\( p = 6 \)[/tex] is a valid solution.
Thus, the exact solution set is [tex]\( \{6\} \)[/tex].
