To find the range, find out what's the maximum and minimum of this function. The natural log function is an increasing function (that is, as x increases from -infinity to +infinity, ln(x) increases). Thus, the maximum that g(x,y,z) can be is ln(25); do you see which this is the case? x^2, y^2, and z^2 are all positive, and they're subtracted from 25. Now, the minimum is a different story; since x, y, and z are unbounded, you can try very large numbers and very small numbers. In fact, it turns out that ln(x) where x<=0 will never work, but ln(x) where x<1 is a negative number. In fact, if you let x^2 + y^2 + z^2 = 24.5, you'll find that g(x,y,z) ~= -0.69. So, let x^2 + y^2 + z^2 = 25-10^-20... that gets you g(x,y,z) ~= -46. Keep trying things; you'll find out that there is no lower bound for g(x,y,z). So your answer is:

( -infinity, ln(25) ]