The half-life of a substance is the time it takes for the quantity of that substance to reduce to half of its original amount. In the equation you provided, f(t) = 249(0.72^t), we can find the half-life by setting f(t) to half of its original value (249/2).
So, we have:
249(0.72^t) = 249/2
Now, we can solve for t:
0.72^t = 1/2
To solve for t, you can take the natural logarithm (ln) of both sides:
ln(0.72^t) = ln(1/2)
Using the properties of logarithms, you can bring down the exponent t in front:
t * ln(0.72) = ln(1/2)
Now, divide both sides by ln(0.72) to isolate t:
t = ln(1/2) / ln(0.72)
Now, you can calculate the value of t:
t ≈ -2.5055
So, the half-life of this substance is approximately -2.5055 units of time. Typically, we express time as positive values, so the half-life is approximately 2.5055 units of time. Please note that the units of time will depend on the context in which the equation is used.