To solve the given exponential equation [tex]\( 3^{(3x - 2)} = 81 \)[/tex], follow these steps:
1. First, express 81 as a power of 3. We know that [tex]\( 81 = 3^4 \)[/tex].
Now the equation becomes:
[tex]\[ 3^{(3x - 2)} = 3^4 \][/tex]
2. Since the bases are the same, we can equate the exponents:
[tex]\[ 3x - 2 = 4 \][/tex]
3. Solve the linear equation [tex]\( 3x - 2 = 4 \)[/tex]:
- Add 2 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 3x = 4 + 2 \][/tex]
[tex]\[ 3x = 6 \][/tex]
- Divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{6}{3} \][/tex]
[tex]\[ x = 2 \][/tex]
So, the solution to the given exponential equation is:
[tex]\[ 3x - 2 = 4 \][/tex]
[tex]\[ x = 2 \][/tex]
