## zmudz one year ago Suppose $$A$$ and $$B$$ are positive real numbers for which $$\log_A B = \log_B A$$. If neither $$A$$ nor $$B$$ is 1 and $$A \neq B$$, find the value of $$AB$$.

let $\log_AB = \log_B A = k$ $$\implies B = A^k ~~\text{and} ~~A = B^k\tag{1}$$ $$\implies AB = A^kB^k=(AB)^k$$ $$\implies AB=1$$ or $$k=1$$ however$$k=1$$ is impossible because it gives $$A=B$$ from $$(1)$$