To solve the equation [tex]\( 9^{x-1} - 2 = 25 \)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
9^{x-1} - 2 = 25
\][/tex]
2. Isolate the term involving the exponent by adding 2 to both sides of the equation:
[tex]\[
9^{x-1} = 27
\][/tex]
3. Recognize that 27 can be written as a power of 3. Recall that [tex]\( 9 = 3^2 \)[/tex] and [tex]\( 27 = 3^3 \)[/tex]. Rewrite the equation using these bases:
[tex]\[
(3^2)^{x-1} = 3^3
\][/tex]
4. Simplify the left-hand side using the properties of exponents [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
3^{2(x-1)} = 3^3
\][/tex]
5. Since the bases are the same, set the exponents equal to each other:
[tex]\[
2(x-1) = 3
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
2x - 2 = 3
\][/tex]
7. Add 2 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
2x = 5
\][/tex]
8. Divide both sides by 2:
[tex]\[
x = \frac{5}{2}
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\( 9^{x-1} - 2 = 25 \)[/tex] is:
[tex]\[
x = \frac{5}{2}
\][/tex]
Given the options:
- [tex]\( x = \frac{1}{2} \)[/tex]
- [tex]\( x = 2 \)[/tex]
- [tex]\( x = \frac{5}{2} \)[/tex]
- [tex]\( x = 4 \)[/tex]
The correct answer is:
[tex]\[
x = \frac{5}{2}
\][/tex]
This corresponds to the third option.
