Certainly! Let's solve the equation step by step:
Given equation:
[tex]\[ 3^{x-1} = 9^{x+2} \][/tex]
First, we recognize that 9 can be written as a power of 3. Specifically:
[tex]\[ 9 = 3^2 \][/tex]
So we can rewrite the right-hand side of the equation:
[tex]\[ 9^{x+2} = (3^2)^{x+2} \][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[ (3^2)^{x+2} = 3^{2(x+2)} \][/tex]
Now our equation looks like:
[tex]\[ 3^{x-1} = 3^{2(x+2)} \][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[ x - 1 = 2(x + 2) \][/tex]
Now let's solve the equation for [tex]\( x \)[/tex]:
[tex]\[ x - 1 = 2x + 4 \][/tex]
To isolate [tex]\( x \)[/tex], first subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ -1 = x + 4 \][/tex]
Then subtract 4 from both sides:
[tex]\[ -1 - 4 = x \][/tex]
[tex]\[ -5 = x \][/tex]
So, the solution is:
[tex]\[ x = -5 \][/tex]
