a) To calculate: A model of the form N(t)=a+b \ln t for the provided data.

a) Given:
The sales of a book tend to increase over the short-term as word-of-mouth makes the book "catch on." The number of bokks sold N(t) for a new novel t weeks after release at a certain book store is given in the table for the first 6 weeks.
$\begin{array}{|cc|}\hline \text{Weeks t}& \text{Number sold N(t)}\\ 1& 20\\ 2& 27\\ 3& 31\\ 4& 35\\ 5& 38\\ 6& 39\\ \hline\end{array}$
Calculation:
Consider the provided table:
$\begin{array}{|cc|}\hline \text{Weeks t}& \text{Number sold N(t)}\\ 1& 20\\ 2& 27\\ 3& 31\\ 4& 35\\ 5& 38\\ 6& 39\\ \hline\end{array}$
Use graphic utility calculator Ti-83 for modeling an equation of provided data:
STEP1: Press STAT key then press 1 in Ti-83 calculator. A table will arise. Fill above values of weeks and corresponding number of books sold in that table.
STEP2: After filling the above data in table press CALC key and then press 0 to select ExpReg to calculate the model function of the provided data.
The regression equation is, ${\displaystyle y=19.68+10.92\ln t}$
The model for the provided data is, ${\displaystyle N\left(t\right)=19.68+10.92\ln t}$,
b) Given:
The number of weeks is, 7.
Formula used:
${\displaystyle N\left(t\right)=19.68+10.92\ln t}$
Calculation:
Consider the model obtained in part (a)
${\displaystyle N\left(t\right)=19.68+10.92\ln t}$
Substitute the value ${\displaystyle t=7}$ in the equation.
${\displaystyle N\left(7\right)=19.68+10.92\ln \left(7\right)}$
${\displaystyle \approx 41}$
The number of books sold for a week 7 would be approximately 41.
c) Given: The statement: "Is it reasonable to assume that this logarithmic trend will continue."
No, the statement is not reasonable because the above logarithmic curve is formed due to sale of book through word of mouth method of publicity which is valid for short term only.
Suppose that negative publicity of book started then the sale of book may go down may not follow logarithmic trend.
Answer: No, the provided statement is not valid.