Question: User: 4 exponent 2x-1 = 64
1. **State the problem:** Solve the equation $$4^{2x-1} = 64$$ for $$x$$.
2. **Recall the formula and rules:** We know that $$a^{m} = a^{n}$$ implies $$m = n$$ if $$a > 0$$ and $$a \neq 1$$.
3. **Rewrite both sides with the same base:**
Since $$4 = 2^2$$ and $$64 = 2^6$$, rewrite the equation as:
$$4^{2x-1} = 64 \Rightarrow (2^2)^{2x-1} = 2^6$$
4. **Simplify the left side using power of a power rule:**
$$(2^2)^{2x-1} = 2^{2(2x-1)} = 2^{4x - 2}$$
So the equation becomes:
$$2^{4x - 2} = 2^6$$
5. **Set the exponents equal:**
$$4x - 2 = 6$$
6. **Solve for $$x$$:**
$$4x - 2 = 6$$
Add 2 to both sides:
$$4x - \cancel{2} + \cancel{2} = 6 + 2$$
$$4x = 8$$
Divide both sides by 4:
$$\frac{\cancel{4}x}{\cancel{4}} = \frac{8}{4}$$
$$x = 2$$
**Final answer:** $$x = 2$$