Given curves $y=3^{x}$ ...(i)
and $y-5^{x}$ ...(ii)
Intersect at the point $(0,1)$.
Now, differentiating Eqs. (i) and (ii) w.r.t. $x$, we get
$\frac{d y}{d x}=3^{x} \log 3$ and $\frac{d y}{d x}=5^{x} \log 5$
$\Rightarrow \left(\frac{d y}{d x}\right)_{(0,1)}=\log 3$ and $\left(\frac{d y}{d x}\right)_{(0,1)}=\log 5$
$\Rightarrow m_{1}=\log 3$ and $m_{2}=\log 5$
Angle between these curves is given by
$\tan \theta=\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}$
$\Rightarrow \tan \theta =\frac{\log 3-\log 5}{1+\log 3 \cdot \log 5}$
$\Rightarrow \theta =\tan ^{-1}\left(\frac{\log 3-\log 5}{1+\log 3 \log 5}\right)$