Question

At what angle $\theta$ do the curves $y=a^x$ and $y=b^x$ intersect $(a\ne b)$?

Answer 1

Let $(x_1,y_1)$ be the point of intersection
$\therefore y_1=a^{x_1}$ ...... (1)
$y_1=b^{x_1}$ ........ (2)
From (1) and (2), we have
$\Rightarrow a^{x_1}=b^{x_1}\Rightarrow x_1=0$ $[\because a\ne b]$
when $x_1=0,y_1=a^0=1$
Hence the point of intersection is $(0,1)$
consider $y=a^x$ and consider $y=b^x$
Diff. w.r.to $x$ and Diff. w.r.to $x$
$\frac{dy}{dx}=a^x\log a$ and $\frac{dy}{dx}=b^x\log b$
$m_1={\left(\frac{dy}{dx}\right)}_{at(0,1)}$ and $m_2={\left(\frac{dy}{dx}\right)}_{at(0,1)}$
$=a^0\log a$ and $=b^0\log b$
$=\log a$ and $=\log b$
Let $\theta$ be the angle between the given curves
$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|=\left|\frac{\log a-\log b}{1+\log a\log b}\right|\Rightarrow \theta$
$={\tan }^{-1}\left[\left|\frac{\log a-\log b}{1+\log a\log b}\right|\right]$