How to prove that $\log(d) , d \sim U(a,b)$ is exponentially distributed (PDF)? I read that logarithm of a standard uniform distribution is exponential PDF. Is it possible to prove that $\log(D)$ with uniform distribution between $(a,b)$ is also a exponential (has exponential PDF)?
In fact, my variable is a distance between distance a and distance b, and I am computing $\log10(D)$ in equal spaces in distance.
Thanks for any help.
Regards.

$X \sim U(a,b)$
Let $Y=\log X $
$F_{Y}(y)=P(Y \le y)=P(\log X \le y)=P(\log X \le y)=P( X \le e^{y})=F_X({e^{y})}$
$\dfrac{d(F_{Y})}{dy}=f_{X}(e^y)e^y=\dfrac{e^y}{b-a}\   \ \bigg( y \in(\log a,\;\log b\bigg)$
Not exponential distribution you can see that.
