If the curve y = a^ x and y = b^ x intersect at angle α , then tanα...

Clearly, the point of intersection of curves is $(0,1)$.

Now, slope of tangent of first curve

${m}_{1} = \dfrac{dy}{dx} = a^{x}loga$

$⇒ \left(\dfrac{dy}{dx}\right)(0,1) = {m}_{1} = loga$

Slope of tangent of second curve,

${m}_{2} = \dfrac{dy}{dx} = b^{x}logb$

$⇒ {m}_{2} = \left(\dfrac{dy}{dx}\right)(0,1) = logb$

$∴ tan\alpha = \left|\dfrac{{m}_{1} - {m}_{2}}{1 + {m}_{1}{m}_{2}}\right| = \dfrac{loga - logb}{1 + logalogb}$