To solve the equation [tex]\(3^{x-5} = 9\)[/tex]:
1. Rewrite the number 9 as a power of 3:
[tex]\(9\)[/tex] can be expressed as [tex]\(3^2\)[/tex], since [tex]\(3^2 = 9\)[/tex].
Therefore, the equation becomes:
[tex]\[
3^{x-5} = 3^2
\][/tex]
2. Equate the exponents:
Since the bases are the same (both bases are 3), we can set the exponents equal to each other:
[tex]\[
x - 5 = 2
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x - 5 = 2
\][/tex]
To isolate [tex]\(x\)[/tex], add 5 to both sides of the equation:
[tex]\[
x = 2 + 5
\][/tex]
Therefore:
[tex]\[
x = 7
\][/tex]
So, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(3^{x-5} = 9\)[/tex] is [tex]\(x = 7\)[/tex].
From the given options:
1. [tex]\( x = -3 \)[/tex]
2. [tex]\( x = 2 \)[/tex]
3. [tex]\( x = 7 \)[/tex]
4. [tex]\( x = 8 \)[/tex]
The correct answer is:
[tex]\[
x = 7
\][/tex]
