Hoa to reverse the exponential expression

ln (natural log) not IN!!!
if y = e^x
=> ln y = ln e^x ( take natural log on both sides )
=> ln y = x ln e ( ln A^n = n ln A)
=> ln y = x ( ln e = 1 )

M(t) = 1 – e^(–0.16t ) = 0.9
e^(–0.16t ) = 0.1
ln e^(–0.16t ) = ln 0.1
(–0.16t )ln e = ln 0.1
–0.16t = ln 0.1
t = –ln 0.1/ 0.16

You can calculate ln by its taylor expansion:
ln(x) = (x-1) - [(x-1)^2]/2 + [(x-1)^3]/3 - [(x-1)^4]/4 + ...
By substituting x = 0.1, you can get the approximation of ln(0.1)

The IN function (on some calculators) can do somthing and then i can use the result which divided by -0.16, I got 14.39, it is "t".

My question is that hoa can I perform the IN function by hand on a paper i.e. doing some + - x / or some other calculations then found "t"

How do I not using calculator for the IN function, i.e. step by step on paper?

乜野意思？你想問乜？