qexp(.5, lam);  10*log(2)
[1] 6.931472
[1] 6.931472

set.seed(1234)           # for reproducibility
m = 10^6;  n = 5;  lam = .1
x = rexp(m*n, lam)
MAT = matrix(x, nrow=m)  # m by n matrix each row a sample
a = rowMeans(MAT)        # vector of m mean
h = apply(MAT,1,median)  # vector of m medians

mean(a); sd(a)
[1] 10.00473
[1] 4.479718

mean(h); sd(h)
[1] 7.833585
[1] 4.625242

est.h = log(2)*a
mean(est.h); sd(est.h)
[1] 6.934753           # aprx mean of MLE = 6.9315
[1] 3.105104           # SD(MLE) < 4.625242