What is the value of \(x\) in the equation \(3^{x+1} = 81\)?

To find the value of (x) in the equation (3^{x+1} = 81), one effective approach is to rewrite the equation by expressing 81 as a power of 3. Knowing that (81 = 3^4), we can substitute this into the equation:

[

3^{x+1} = 3^4

]

Since the bases are the same (both are base 3), we can set the exponents equal to each other:

[

x + 1 = 4

]

To isolate (x), we subtract 1 from both sides of the equation:

[

x = 4 - 1

]

This simplifies to:

[

x = 3

]

Thus, the value of (x) is indeed 3. This calculation shows that when the equation is simplified by expressing 81 in terms of base 3, we find the exponent directly, demonstrating that understanding exponential relationships is critical in problems involving powers.