To solve the equation [tex]\(9^{x-3} = 729\)[/tex], let's follow these steps:
1. Recognize the base relationships:
- Notice that 9 can be written as [tex]\(3^2\)[/tex].
- Notice that 729 can be written as [tex]\(3^6\)[/tex].
2. Rewrite the equation using powers of 3:
[tex]\[
9^{x-3} = 729
\][/tex]
[tex]\[
(3^2)^{x-3} = 3^6
\][/tex]
3. Simplify the left-hand side:
When we raise a power to another power in exponents, we multiply the exponents.
[tex]\[
3^{2(x-3)} = 3^6
\][/tex]
4. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other.
[tex]\[
2(x-3) = 6
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Distribute the 2 on the left-hand side:
[tex]\[
2x - 6 = 6
\][/tex]
- Add 6 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x = 6 + 6
\][/tex]
[tex]\[
2x = 12
\][/tex]
- Divide by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{12}{2}
\][/tex]
[tex]\[
x = 6
\][/tex]
So, the solution to the equation [tex]\(9^{x-3} = 729\)[/tex] is:
[tex]\[
x = 6
\][/tex]
