Prove/Disprove: $f(t),g(t)$ are linearly indep. implies $\ln |f(t)|,\ln|g(t)|$ are linearly indep.

Question

Given that $f(t),g(t)$ are linearly independent functions does this imply that $\ln|f(t)|,\ln|g(t)|$ are linearly independent as well?

The idea seems logically sound since $F: f(t)\rightarrow \ln|f(t)|$ and $G: g(t)\rightarrow \ln|g(t)|$ are both inductive. However I am having trouble conceptualizing a proof.

If you have any ideas on starting the proof or counterexample I'd appreciate it.

Answer 1

This is not true: $e^x$ and $e^{2x}$ are linearly independent, while $x, 2x$ are linear dependent.

Answer 2

The $ln$'s aren't necessarily independent. Take, for example, $f(x)=x$ and $g(x)=x^2$. These two functions are linearly independent. However, their logarithms are dependent:
ln$|f(x)| = \text{ln}|x|$, and ln$|g(x)|=\text{ln}|x^2|=2\text{ln}|x|$ are obviously dependent functions.