If $f:R\to R$ defined by $f\left(x\right)=\{\begin{array}{c}{a}^{2}co{s}^{2}x+b^{2}si{n}^{2}x, x\le 0\\ e^{ax+b}, x>0\end{array}$ is continuous then

$f\left(0\right)=\underset{x\text{}\to 0^{-}}{Lt}f\left(x\right)=\underset{x\to 0^{-}}{Lt}f\left(x\right)$

$a^2=\underset{x\to 0^{-}}{Lt}\left(a^2{\cos }^{2}x+b^2{\sin }^{2}x\right)=\underset{x\to 0^{+}}{Lt}{e}^{ax+b}$

$\therefore a^2=e^b$ $\Rightarrow b={\log }_e$

$a^2\Rightarrow b=2{\log }_e\left|a\right|$