To solve the given exponential equation [tex]\(3^{(3x-2)}=81\)[/tex], we start by recognizing that 81 can be written as a power of 3. Specifically, [tex]\(81 = 3^4\)[/tex].
This transforms the given equation into:
[tex]\[3^{(3x-2)}=3^4\][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[3x - 2 = 4\][/tex]
To solve for [tex]\(x\)[/tex], we now solve the linear equation:
[tex]\[3x - 2 = 4\][/tex]
First, add 2 to both sides of the equation:
[tex]\[3x - 2 + 2 = 4 + 2\][/tex]
[tex]\[3x = 6\][/tex]
Next, divide both sides by 3:
[tex]\[x = \frac{6}{3}\][/tex]
[tex]\[x = 2\][/tex]
Therefore, the solution is:
[tex]\[3x - 2 = 4\][/tex]
and
[tex]\[x = 2\][/tex]
