To determine which equation is equivalent to [tex]\( 9^{x-3} = 729 \)[/tex], let's rewrite the equation in different forms and analyze the options given.
First, we need to express [tex]\( 729 \)[/tex] as a power of [tex]\( 9 \)[/tex]:
[tex]\[
9^{x-3} = 729
\][/tex]
To find this power, note that [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex]:
[tex]\[
9 = 3^2
\][/tex]
Then [tex]\( 9 \)[/tex] raised to any power [tex]\( y \)[/tex] can be written as:
[tex]\[
9^y = (3^2)^y = 3^{2y}
\][/tex]
Thus, [tex]\( 9^{x-3} \)[/tex] can be rewritten as:
[tex]\[
9^{x-3} = (3^2)^{x-3} = 3^{2(x-3)}
\][/tex]
We need to express [tex]\( 729 \)[/tex] as a power of [tex]\( 3 \)[/tex]:
[tex]\[
729 = 3^6
\][/tex]
So the equation becomes:
[tex]\[
3^{2(x-3)} = 3^6
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
2(x-3) = 6
\][/tex]
Simplify the equation:
[tex]\[
2x - 6 = 6
\][/tex]
[tex]\[
2x = 12
\][/tex]
[tex]\[
x = 6
\][/tex]
Now, let’s go back to the options to find the one that matches this transformed equation:
1. [tex]\( 9^{x-3} = 9^{81} \)[/tex]
2. [tex]\( 9^{x-3} = 9^3 \)[/tex]
3. [tex]\( 3^{x-3} = 3^6 \)[/tex]
4. [tex]\( 3^{2(x-3)} = 3^6 \)[/tex]
Option (2) reads [tex]\( 9^{x-3} = 9^3 \)[/tex].
By comparing it with [tex]\( 9^{x-3} = 729 \)[/tex], and since we found [tex]\( 729 = 9^3 \)[/tex], this option is correct.
Therefore, the equation equivalent to [tex]\( 9^{x-3} = 729 \)[/tex] is:
[tex]\[
9^{x-3} = 9^3
\][/tex]
