Find the value of x satisfying x^ (x+3) = 7 — JEE Mathematics

By the fundamental property of logarithms, $b^{\log_b(a)} = a$ provided $a > 0, b > 0, b \ne 1$.
Here, the base of both the expression and the logarithm is $x$.
Thus, $x^{\log_x(x+3)} = x+3$.

The equation becomes:

$$x + 3 = 7 \implies x = 4$$

We must verify constraints:

• Argument $x + 3 = 4 + 3 = 7 > 0$ (Valid)
• Base $x = 4 > 0$ and $x \ne 1$ (Valid)

Answer: 4