Do I need to log transform these two kinds of variables? Before regression (like fixed-effects, random-effects), for these two kinds of variables:

1) Price levels (e.g. price level of capital formation)
They are usually ratios like 0.987, 1.022..., based on USA 2010=1

2) % of GDP (e.g. broad money as % of GDP)
They are also ratios (%) like 80.5, 99.7 (or in decimal form 0.805, 0.997)

Do I need to log transform them, and how can I log transform them?
You don't need to log transform these variables. Consider the following model


  $\log y = \beta x.$


Then, you can interpret $\beta$ as a semi-elasticity, i.e., the percentage change in $y$ when $x$ increases by 1 unit. 

To see this you can transform the previous equation as: 
$y = \exp(\beta x),$ in which case $dy/dx = \beta \exp(\beta x)$. Thus, the percentage change in $y$ when $x$ changes is $\frac{dy/dx}{y} = \frac{\beta\exp(\beta x)}{\exp(\beta x)} = \beta$.

Note also that 


  $\Delta \log y = \beta \Delta x$ so $100.\Delta \log y = (100.\beta) \Delta x$


It follows that 


  $\% \Delta y \approx (100.\beta) \Delta x$