slope of a linear function in semi-log plot

Question

I have a decreasing linear function. So,the slope of this function df/dt <0. Now, if we plot this function in a semi-log plot with log(t) in horizontal axis and y in vertical axis, can we say that this is still a decreasing function with df/d(logt)<0?

Answer 1

Yes. $s\le t\Rightarrow\log(s)\le \log(t)\Rightarrow f(\log(s)) \ge f(\log(t))$, since $\log$ is monotonically increasing and $f$ is monotonically decreasing.

Answer 2

Consider $$\frac{dy}{dt}=\frac{dy}{d\log(t)}\times \frac{d\log(t)}{dt}=\frac 1t\frac{dy}{d\log(t)}$$ $$\frac{dy}{d\log(t)}=t\frac{dy}{dt}$$ Then $\cdots$