Set-and-forget betting requires logarithmic utility

We show that among smooth concave utilities satisfying the Inada conditions, the logarithm is the only one for which the optimal sequential strategy does not depend on the horizon. We call a sequential strategy set-and-forget if the same value process is optimal at every horizon, against every positive-martingale evidence stream, without re-solving the dual problem. The growth-rate-optimal (GRO) $e$-variable, the maximizer of the expected log, is therefore the unique one whose optimality is preserved under arbitrary truncation of the data stream, so a statistician who uses the log score need not know the sample size in advance. The proof shows that the perspective function $\psi(r)=rI(r)$, with $I=(U')^{-1}$, must be affine under every multiplicative mean-preserving spread, and the Inada conditions then force $U(x)=a\log x+b$. The same argument shows that the growth-rate optimum, the numéraire of the alternative class, and the reverse relative-entropy projection coincide only under the logarithm. For any non-log utility an explicit two-point mean-one martingale witnesses strict suboptimality. On real anytime-valid streams the log-optimal $e$-process accumulates evidence where the non-log scores lag or stall.