Half life (mathematically #T_(1/2)#) is how long it takes for half of the atoms in a substance to radioactively decay.
If you want to know the maths behind their relationship,
#N = N_0e^(-lambdat)# applies to radioactive substances, where
#N# is the number of radioactive atoms at time #t#
#N_0# is the number of radioactive atoms at the beginning of the process, when #t = 0#
#e# is Euler's constant, #approx 2.71828#
#t#, as mentioned, is time, and
#lamda# is the decay rate, which is a constant value for each isotope. It can also be thought of as the probability for an atom to decay in a unit time.
When #t = T_(1/2)#, then half the initial atoms have decayed, which means that #N = N_0/2#.
Substituting this into the equation,
#N_0/2 = N_0e^(-lambdaT_(1/2))#
#1/2 = e^(-lambdaT_(1/2))#
Taking natural logs and rearranging from there,
#ln(1/2) = -lambdaT_(1/2)#
#ln1 - ln2 = -ln2 = -lambdaT_(1/2)#
#ln2 = lambdaT_(1/2)#
#T_(1/2) = ln2/lambda = 0.693/lambda#
which is mathematically how the rate of radioactive decay is related to half life.
From this equation, we can see that if decay rate (#lambda#) increases, #T_(1/2)# will get shorter, and if decay slows down, half life will increase.
They are inversely proportional.
