Question: 2 – Solving an Exponential Equation
LEARNING OBJECTIVE: Solve an exponential equation by rewriting the base.
If $25^{x+1} = 5^{-x+11}$, then $x$ equals __________.
a.) 7
b.) 9
c.) 3
d.) 5
1. **State the problem:** Solve the exponential equation $$25^{x+1} = 5^{-x+11}$$ for $x$.
2. **Rewrite the bases:** Note that $25$ and $5$ are related since $25 = 5^2$. So rewrite $25^{x+1}$ as $(5^2)^{x+1}$.
3. **Apply the power of a power rule:** $$(5^2)^{x+1} = 5^{2(x+1)} = 5^{2x+2}$$
4. **Rewrite the equation with the same base:**
$$5^{2x+2} = 5^{-x+11}$$
5. **Since the bases are the same and the function $5^y$ is one-to-one, set the exponents equal:**
$$2x + 2 = -x + 11$$
6. **Solve for $x$:**
Add $x$ to both sides:
$$2x + x + 2 = 11$$
$$3x + 2 = 11$$
Subtract $2$ from both sides:
$$3x = 11 - 2$$
$$3x = 9$$
Divide both sides by $3$:
$$x = \frac{9}{3}$$
$$x = 3$$
7. **Final answer:** $x = 3$ which corresponds to option c.) 3.