We have the fourier-transform:
$$F\{f\}(w) = \int_{-\infty}^\infty f(x)\exp(iwx)dx$$
Which has extremely many applications and interpretations throughout science and engineering. For instance since it has the famous convolution rule: $$F(f*g) = F(f)\cdot F(g)$$ which can be used to speed up convolutions, it also has special differentiation and integration rules: $$F\left\{\frac{\partial f}{\partial x}\right\}(w) = wiF\{f\}(w)$$ $$F\left\{\int f\right\}(w) = \frac{F\{f\}(w)}{iw}$$ making it very useful in solving differential equations.
So I have thought about what happens if we replace the complex exponential with a logarithm:
$$L\{f\}(w) = \int_{0}^\infty f(x)\log(wx)dx$$
Do you think this would make sense to investigate? Could it have any interesting properties? Are there any other integral transforms involving logarithms which are interesting or useful?
EDIT Observation by Tom-Tom, we can rewrite it as:
$$\int_0^\infty f(x)\log(\omega x)\mathrm dx= \left(\int_0^\infty f(x)\mathrm dx\right)\log(\omega) +\int_0^\infty f(x)\log(x)\mathrm dx = a \log(w) + b$$ with constants: $a$ is the integral of $f$ and $b$ is a log-weighted integral of $f$.
