To solve the equation [tex]\( 3^{x-5} = 9 \)[/tex], we can follow these steps:
1. Recognize that 9 can be written as a power of 3. Specifically, [tex]\( 9 = 3^2 \)[/tex].
2. Rewriting the equation [tex]\( 3^{x-5} = 9 \)[/tex] in terms of the same base gives:
[tex]\[
3^{x-5} = 3^2
\][/tex]
3. Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
x - 5 = 2
\][/tex]
4. Solve for [tex]\( x \)[/tex] by adding 5 to both sides of the equation:
[tex]\[
x - 5 + 5 = 2 + 5
\][/tex]
[tex]\[
x = 7
\][/tex]
The solution to the equation [tex]\( 3^{x-5} = 9 \)[/tex] is [tex]\( x = 7 \)[/tex]. Therefore, the correct answer is:
[tex]\[ x = 7 \][/tex]
