Continuous function $g$ satisfying $g(x + y) = 5g(x)g(y)$

Question

Let $g$ be a continuous function with $g(1) = 1$ such that $$g(x + y) = 5g(x)g(y)$$ for all $x$, $y$. Find $g(x)$.

Answer 1

setting $y=1$ gets

$$g(x+1)=5g(x)g(1)=5g(x)$$

So every time you increase the argument by $1$, you multiply by $5$. Can you see what function has this property? and how to prove that the solution is unique? You probably won't get complete answers until you post some of your work, so we know what specifically to help you with.

Answer 2

A small beginning: Let $g(t)=\frac{h(t)}{5}$. Then our functional equation can be rewritten as $$\frac{h(x+y)}{5}=5\frac{h(x)}{5}\frac{h(y)}{5},$$ or equivalently $$h(x+y)=h(x)h(y).$$ This one is a standard functional equation that has been discussed on this site and elsewhere.