Question

if many trees are removed from a forest by logging, what is the most immediate effect on the carbon cycle in that forest

Answer 1

as we know plants take carbon dioxide and realese 02 so if trees are cut co2 level of that forest wil also increase causing global warming

Answer 2

Most immediate effect on the carbon cycle in that forest:

Large scale deforestation in an area can have adverse effects on the carbon cycle by increasing the amount of $CO_2$ in the immediate area. Trees play a major role in the regulation of $CO_2$ in the atmosphere, a sudden loss of many trees would first affect the balance of $O_2$ and $CO_2$.  

Trees which would have otherwise decayed and become a part of the carbon cycle would be lost, but the immediate impact of deforestation would be to the control of $CO_2$ in the atmosphere.