To solve the equation [tex]\( 3^{-x} = 9 \)[/tex], follow these steps:
1. Recognize the Bases:
The number 9 can be rewritten as a power of 3. Since [tex]\( 9 = 3^2 \)[/tex], we can replace 9 in the equation with [tex]\( 3^2 \)[/tex].
So the equation becomes:
[tex]\[
3^{-x} = 3^2
\][/tex]
2. Set the Exponents Equal to Each Other:
Since the bases are the same (both are 3), we can set the exponents equal to each other. This gives us a simpler linear equation to solve:
[tex]\[
-x = 2
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], we need to isolate [tex]\( x \)[/tex] by multiplying both sides of the equation by -1:
[tex]\[
x = -2
\][/tex]
Thus, the solution to the equation [tex]\( 3^{-x} = 9 \)[/tex] is:
[tex]\[
x = -2
\][/tex]
