1. The problem statement, all variables and given/known data This is for a first order step function response, where T is the time constant and x(t) is H...(input). For settling time, the percentage most often taken is 1% within the final value, which gives the settling time, ts, from: 0.99H = H(1 - e-ts/T) We get, ts = 4.6T. Now, in some applications it may be appropriate to take another value of output. For example: time to get within 0.1% of final value. I saw this example somewhere but I don't understand their working method which was: t0.1% = t1% + ln(10)T I understand that the time to get within 0.1% of the final value will be "the time to get within 1% of the final value (H)" + something. But I don't understand why it's plus ln(10)T. It's been bothering me since last night and I need to know why they did that and still got the right answer of 6.9T. When you do it the normal way, you get 6.9T too. ie: 0.999H = H(1 - e-ts/T) e-ts/T = 0.001 ts = -Tln(0.001) ts = Tln(1000) ts = 6.9T t0.1% = 6.9T My question is, why did they add "ln(10)T" to "the time it takes to get within 1% of the final value"? I have a feeling I might just be overlooking something small. 2. Relevant equations y(t) = H(1 - e-ts/T) 3. The attempt at a solution At first I thought that the ln(10)T was the time it takes to cover 0.009H, because: 0.999H = 0.99H + 0.009H, which is: t0.1% = t1% - ln(0.991)T t0.1% = t1% + ln(1.01)T But I guess I was wrong because clearly ln(10)T is not equal to ln(1.01)T.