Let f: R arrow R be a continuous function, which satisfies...

From given integral equation, $f(0)=0$. Also, differentiating the given integral equation w.r.t. $x$, we get \begin{gathered} f^{\prime}(x)=f(x) \\ \text{ If } \quad f(x) ≠ 0 \Rightarrow \frac{f^{\prime}(x)}{f(x)}=1 \\ \Rightarrow \quad \log f(x)=x+C \Rightarrow f(x)=e^C e^x \\ \because f(0)=0 \Rightarrow e^C=0, \text{ a contradiction }\\ \therefore f(x)=0, \forall x \in R \Rightarrow f(\ln 5)=0 \end{gathered}