We have: #- e^(- 3.9 n - 1) - 1 = - 3#
Multiplying both sides of the equation by #- 1#:
#Rightarrow - 1 (- e^(- 3.9 n - 1) - 1) = - 1 times - 3#
#Rightarrow e^(- 3.9 n - 1) + 1 = 3#
Subtracting #1# from both sides:
#Rightarrow e^(- 3.9 n - 1) + 1 - 1 = 3 - 1#
#Rightarrow e^(- 3.9 n - 1) = 2#
Applying #ln# to both sides:
#Rightarrow ln(e^(- 3.9 n - 1)) = ln(2)#
Using the laws of logarithms:
#Rightarrow (- 3.9 n - 1)(ln(e)) = ln(2)#
#Rightarrow (- 3.9 n - 1) times 1 = ln(2)#
#Rightarrow - 3.9 n - 1 = ln(2)#
Adding #1# to both sides:
#Rightarrow - 3.9 n - 1 + 1 = ln(2) + 1#
#Rightarrow - 3.9 n = ln(2) + 1#
Dividing both sides by #- 3.9#:
#Rightarrow frac(- 3.9 n)(- 3.9) = frac(ln(2) + 1)(- 3.9)#
#therefore n = - frac(ln(2) + 1)(3.9)#
