Question: Find the solution to the following equation.
$5^{2x} = 25^{4 - x}$
$x = 2$
$x = \frac{20}{7}$
$x = \frac{1}{2}$
$x = \frac{7}{20}$
1. **State the problem:** Solve the equation $$5^{2x} = 25^{4 - x}$$ for $x$.
2. **Rewrite the bases:** Note that $25 = 5^2$, so rewrite the right side:
$$5^{2x} = (5^2)^{4 - x}$$
3. **Apply the power of a power rule:**
$$(5^2)^{4 - x} = 5^{2(4 - x)} = 5^{8 - 2x}$$
4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal:
$$2x = 8 - 2x$$
5. **Solve for $x$:**
Add $2x$ to both sides:
$$2x + 2x = 8 - 2x + 2x$$
$$4x = 8$$
Divide both sides by 4:
$$\cancel{4}x = \frac{8}{\cancel{4}}$$
$$x = 2$$
6. **Check the answer:** Substitute $x=2$ back into the original equation:
Left side: $$5^{2(2)} = 5^4 = 625$$
Right side: $$25^{4 - 2} = 25^2 = 625$$
Both sides equal, so $x=2$ is correct.
**Final answer:** $$x = 2$$