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# proving existence of continuous function (complicated with ln)

me is trying to answer this complicate question.

question is: let h(x) be continuous function on I that contains $x=0$ and $h(0)=0, h(x)>-1, h(x) \neq 0$.

f(x) is define be $f(x)=(\frac{(1+h(x))^\frac{1}{h(x)}}{e})^\frac{1}{x}$ for every $0 \neq x \in I$

show that exist continuous function g(x) on I so that $g(0)=-\frac{1}{2}$ and $f(x)=exp(\frac{h(x)}{x}g(x))$.

so me try do compare $(\frac{(1+h(x))^\frac{1}{h(x)}}{e})^\frac{1}{x}=exp(\frac{h(x)}{x}g(x))$ (because f(x)=f(x)) and use ln(f(x)),to isolate $g(x)$ but me get lost. me research question and know that because $h(x)$ continuous than $f(x)=exp(\frac{h(x)}{x}g(x))$ is not continuous because of hole in x=0 even altough h(x) continuous in x=0. so me try isolate g(x) and get lost. me also try put f(0) to show $g(0)=-\frac{1}{2}$ but don't go nowhere.

can you please show me how you solve this hard problem? thank you helping me.