### Instruction: Solve for $x$: $2^{x-3}=4^{x+1}$.### Response: I notice that both sides of the equation have bases that are powers of 2, so I can rewrite them using the same base.
Using the exponent rules, I get $2^{x-3}=2^{2(x+1)}$.
Now that both sides have the same base, I can equate the exponents and solve for $x$.
I get $x-3=2(x+1)$, which simplifies to $x-3=2x+2$.
Subtracting $x$ from both sides, I get $-3=x+2$.
Subtracting 2 from both sides, I get $-5=x$.
This is the solution for $x$.