If $$\log_4 x = a$$ and $$\log_{25} x = b$$, then $$\log_x 10$$ is

Since $$\log_ 4 x =a $$ and $$\log_{25} x =b$$, we have $$x= 4^a = 25^b$$

Therefore, $$4= x^{\dfrac{1}{a}}$$ and $$25=x^{\dfrac{1}{b}}$$. Lastly, $$100 = 4\times 25= x^{\dfrac{1}{a}}\times x^{\dfrac{1}{b}} = x^{\dfrac{a+b}{ab}}$$

Finally, since $$100=10^{2}$$, we get $$10=100^{\dfrac{1}{2}}$$ or $$10 = x^{\dfrac{a+b}{ab}\times \dfrac{1}{2}} = x^{\dfrac{a+b}{2ab}}$$

Therefore, we get $$\log_x {10} = \dfrac{a+b}{2ab}$$