Question

If $5^{x-1} = 25^{y+1}$, what is the value of $x$ in terms of $y$?

• A. $2y + 3$
• B. $2y + 2$
• C. $2y + 1$
• D. $2y$

Hint:

Express all terms with the same base and equate exponents to solve.

Answer 1

The Correct Option is A

- Step 1: Rewrite: $25 = 5^2$, so $25^{y+1} = (5^2)^{y+1} = 5^{2(y+1)} = 5^{2y + 2}$.
- Step 2: Given: $5^{x-1} = 5^{2y + 2}$.
- Step 3: Equate exponents: $x - 1 = 2y + 2 \implies x = 2y + 3$.
- Step 4: Verify: If $y = 0$, $25^1 = 25$, $5^{x-1} = 25 \implies 5^{x-1} = 5^2 \implies x - 1 = 2 \implies x = 3$. Check: $x = 2 \times 0 + 3 = 3$.
- Step 5: Option (1) is $2y + 3$, correct.