AIME 2023a Problem 2

Positive real numbers $b \not= 1$ and $n$ satisfy the equations $$\sqrt{\log_b n} = \log_b \sqrt{n} \text{and} b \cdot \log_b n = \log_b (bn).$$ The value of $n$ is $\frac{j}{k},$ where $j$ and $k$ are relatively prime positive integers. Find $j+k.$