Define the function $\mathbb{\text{⨍ : ℝ→ℝ}}$ by

$f\left(x\right)=\left\{\begin{array}{l}{\left(1+3x\right)}^{\frac{4}{3}} if x\ne 0\\ a if x=0\end{array}\right.$

If $\text{f}\left(x\right)$ is continuous. then $\log a=$

The given funciton is $f\left(x\right)=\left\{\begin{array}{l}{\left(1+3x\right)}^{\frac{4}{3}} if x\ne 0\\ a if x=0\end{array}\right.$

Given the function is continous every where

 so the function is continuous at $x=0$

hence, $\underset{x\to 0}{Lim} f\left(x\right)=f\left(0\right)$

We know that $\underset{x\to 0}{Lim} {\left(1+ax\right)}^{\frac{b}{x}}=e^{\frac{a}{b}}$

Given $f\left(0\right)=a$

$a=\underset{x\to 0}{Lim} {\left(1+3x\right)}^{\frac{4}{3x}}=e^4$