Relation of two definite integrals For a differentiable continuous function $g(u)$ over a finite region $R$, I know that
$$\int_R e^u g\; u\; du = 1$$
Is there any way to determine from this the value of
$$\int_R 10^u g\; u\; du$$
The second integral is

$$J=\int 10^u g(u)du =$$

$$=\int e^{uln(10)}g(u)du=.$$

put $v=u ln(10)$

then

$$J=\frac{1}{ln(10)}\int e^v g(\frac{v}{ln(10)})dv.$$

So, in general, we cannot get tha value of $J$

from $I=\int e^u g(u)du=1$.

if $g$ is linear  then $J=\frac{1}{(ln(10))^2}$.
No, it depends on $g$ look at the two cases $g(x)=x$ and $g(x)=x^2$.