Tangent of the angle at which the curve $y=a^x$ and $y=b^x$ $(a\ne b>0)$ intersect is given by

Intersection of the two curves is given by $a^x$ $=b^x$  which implies $x = 0$ If $\alpha$ is the angle at which the two curves intersect then 

$\tan \alpha =\frac{m_1-m_2}{1+m_1{m}_2}={\left.\frac{a^x\log a-b^x\log b}{1+a^x{b}^x\left(\log a\right)\left(\log b\right)}\right|}_{x=0}$

$=\frac{(\log a/b)}{1+\left(\log a\right)\left(\log b\right)}$                    (Putting $x=0)$