We show how a log-Sobolev inequality can be manipulated to give an improved version of itself. There is a very simple manipulation of some log-Sobolev inequalities by means of which the inequality is strengthened, leading in turn to inequalities which could not have been derived from the unimproved version. We begin with a particularly simple illustration of the manipulation, and will then give a slight generalization to other settings. In order to keep things relatively simple, we will generally work in the smooth (infinitely differentiable) category. Thus we begin with a smooth compact Riemannian manifold, metric scaled so total volume is one, with gradient V and (positive) Laplacian A. Assume we have a defective logarithmic-Sobolev inequality (LSI for short): T JVff12 > Jf2 In f2 _ f2 In Jf2 _ p J f2 where f is a real smooth function, and p (p > 0) is the defect. As is well known, the above implies T JiVfi2 > Jf2v Jf2In Jev p Jf2 for any smooth V, and the modified inequality holding for all such V implies, by a simple approximation argument, the original LSI. The advantage of the modified inequality is its simple quadratic character. In it we may replace f by f + x; the left-hand side does not change, so we choose x to maximize the right-hand side. Simple algebra yields the inequality: TJ~vfI2 ? f ~ ff~ln~eV p~f2+(fflIn feV + pfff fV) 2 TJ Vf 12 > f2v _ f 2 In JeV _ p f f2 + JfIn e+piV) I ] ] l~~~~~~nf eV +p f V Since p > 0, the denominator of the last term on the right is positive if V is not constant. If f is not constant, we may now set V = ln f2 (or more properly let V approach lnf2 through a sequence of smooth functions) to get an inequality obviously superior to the original. Received by the editors July 3, 1996. 1991 Mathematics Subject Classification. Primary 47D07, 58G11.