To solve the equation [tex]\( 27 = 9^{x-1} \)[/tex], let's proceed through the steps in a detailed manner:
1. Express 27 and 9 as powers of the same base:
We know that 27 can be written as [tex]\( 3^3 \)[/tex] and 9 can be written as [tex]\( 3^2 \)[/tex].
So we can rewrite the equation:
[tex]\[
3^3 = (3^2)^{x-1}
\][/tex]
2. Simplify the right side of the equation:
Using the property of exponents, [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify the right-hand side:
[tex]\[
(3^2)^{x-1} = 3^{2(x-1)}
\][/tex]
Thus, the equation becomes:
[tex]\[
3^3 = 3^{2(x-1)}
\][/tex]
3. Equate the exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
3 = 2(x-1)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
a. First, distribute the 2 on the right side:
[tex]\[
3 = 2x - 2
\][/tex]
b. Add 2 to both sides of the equation to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
3 + 2 = 2x
\][/tex]
[tex]\[
5 = 2x
\][/tex]
c. Finally, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5}{2}
\][/tex]
[tex]\[
x = 2.5
\][/tex]
Thus, the solution to the equation [tex]\( 27 = 9^{x-1} \)[/tex] is:
[tex]\[
x = 2.5
\][/tex]
