- Suppose two variables x and y have no linear correlation. If we transform the data by replacing each y value with its base-10 logarithm, then will x and log $y$ also have zero correlation?
- In transforming a power function, does the base $c$ for log$_c$ $y$ and for log$_c$ $x$ matter? Can any $c$ greater than 0 be used?
No -- the correlation can be non-zero.
Here's an example in R:
x = (0:10) + 0.003 y = 1 + (x-mean(x))^2 cor(x,y)  -9.892917e-18 cor(log(x),log(y))  -0.379543
x = runif(1000000) y = runif(1000000,0,sqrt(.25-(x-.5)^2))
Note that if $X$ and $Y$ are independent (and I assume positive, so you can take logs), then $\log(X)$ and $\log(Y)$ will also be independent, but in general it's not the case that you can take logs of uncorrelated variables and still have them uncorrelated.
- Changing the base of the log only changes the result by a constant scaling factor. So anything that is unaffected by changing the scale will not be affected by changing the base of the logarithm.