2. \( 3^{4 x+1}-5=22 \)

To solve the equation \( 3^{4x+1} - 5 = 22 \), first, add 5 to both sides to get \( 3^{4x+1} = 27 \). Now notice that 27 can be rewritten as \( 3^3 \). Therefore, we can set the exponents equal to each other: \( 4x + 1 = 3 \). Subtracting 1 from both sides gives \( 4x = 2 \), and dividing by 4 yields \( x = \frac{1}{2} \). Now, let's have some fun with exponential equations! Did you know that exponentials are everywhere in real life? From population growth to bank interest, understanding how to manipulate these equations can give you insights into various fields including finance, biology, and technology. It's like having a secret power! Also, keep an eye out for common pitfalls! When you're dealing with exponents, it's easy to forget about the base. Always ensure that your bases are the same when equating them. Missteps in manipulating exponent rules can lead to wild goose chases in solving problems! So, double-check your work for a smoother solving experience.