Variables t and N are such that when lgN is plotted against lg t, a straight line graph passed through the points (0.45,1.2) and (1, 3.4) is obtained

(i) Express the equation of the straight line graph in the form lgN = m lg t + lg c, where m and c are constants to be found.

(ii) Hence express N in terms of t

The expression for N in terms of t is N = k tn.

Given that the variables t and N are such that when lgN is plotted against lg t, a straight line graph passed through the points (0.45,1.2) and (1, 3.4) is obtained. The question requires us to express N in terms of t.
A straight line graph represents an equation in the form y = mx + c, where y represents the dependent variable, m represents the slope of the line, x represents the independent variable, and c represents the y-intercept. Thus, we can write the equation of the line obtained as:
lg N = m lg t + c (1)
To find the value of m and c, we can use the two points (0.45, 1.2) and (1, 3.4) that the line passes through. Substituting the values of x and y into equation (1), we get:
lg N1 = m lg t1 + c   ...(2)
lg N2 = m lg t2 + c   ...(3)
where N1 = antilog(1.2), t1 = antilog(0.45), N2 = antilog(3.4), and t2 = antilog(1).
Taking the logarithm of both sides of equation (2) gives:
lg lg N1 = lg(m lg t1 + c)
Plotting a graph of lg lg N1 against lg t1 gives a straight line with a slope of m and a y-intercept of lg c. Similarly, from equation (3), we can plot a graph of lg lg N2 against lg t2 to obtain another straight line with a slope of m and a y-intercept of lg c.
Substituting the values of N1, t1, N2, and t2 into equations (2) and (3), we get:
1.2 = mt1 + c   ...(4)
3.4 = mt2 + c   ...(5)
Subtracting equation (4) from equation (5) eliminates c and gives:
2.2 = m(t2 - t1)
Simplifying the above equation gives:
m = 2.2 / (t2 - t1)
Substituting the value of m in equation (4) gives:
1.2 = [2.2 / (t2 - t1)]t1 + c
Simplifying the above equation gives:
c = 1.2 - [2.2t1 / (t2 - t1)]
Therefore, the equation of the line is:
lg N = [2.2 / (t2 - t1)] lg t + [1.2 - 2.2t1 / (t2 - t1)] (6)
We need to express N in terms of t. Taking antilogarithm of both sides of equation (6) gives:
N = at[2.2 / (t2 - t1)] x at[1.2 - 2.2t1 / (t2 - t1)]
Simplifying the above equation gives:
N = k tn
where k = at[1.2 - 2.2t1 / (t2 - t1)] and n = 2.2 / (t2 - t1).

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can someone help answer this and explain how you did it