## Generate uniform data on (0,1).  The expected value is 1/2.
X = runif(1e4)

## Simulate the exact result for the log transform.
L1 = var(log(X))

## The mathematical approximation for the log function.
L2 = 1/3 

## Simulate the exact result for the squaring transform.
S1 = var(X^2)

## The mathematical approximation for the squaring function.
S2 = 1/12