Sure, let's solve the equation [tex]\( 3^{x-2} = 3^7 \)[/tex].
Step-by-step:
1. Observe the Equation: We are given that [tex]\( 3^{x-2} = 3^7 \)[/tex].
2. Equal Bases: Since the bases of the exponentials on both sides of the equation are the same (both are base 3), we can equate the exponents directly.
3. Set the Exponents Equal:
[tex]\[
x - 2 = 7
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x - 2 = 7
\][/tex]
To isolate [tex]\( x \)[/tex], add 2 to both sides of the equation:
[tex]\[
x - 2 + 2 = 7 + 2
\][/tex]
Simplifying, we get:
[tex]\[
x = 9
\][/tex]
Hence, the solution to the equation [tex]\( 3^{x-2} = 3^7 \)[/tex] is [tex]\( x = 9 \)[/tex].
Therefore, the correct answer is [tex]\( \boxed{9} \)[/tex].
