Exponent and Logarithms
1. (a) The function f is defined by
(i) Find the minimum value of f.
(2)
(ii) Prove that ex ³ 1 + x for all real values of x.
(3)
(b) Use the principle of mathematical induction to prove that
for all integers n ³ 1.
(6)
(c) Use the results of parts (a) and (b) to prove that
> n.
(4)
(d) Find a value of n for which
> 100
(3)
(Total 18 marks)
2. Find , giving the answer in the form a ln 2, where a Î .
Working: 


Answer: .......................................................................... 
(Total 6 marks)
3. Find the exact value of x satisfying the equation
(3x)(42x+1) = 6x+2.
Give your answer in the form where a, b Î .
Working: 


Answer: ......................................................................... 
(Total 6 marks)
4. Solve log16 .
Working: 


Answer: ......................................................................... 
(Total 6 marks)
5. Solve 2(5x+1) = 1 + , giving the answer in the form a + log5 b, where a, b Î .
Working: 


Answer: ......................................................................... 
(Total 6 marks)
6. Find an expression for the sum of the first 35 terms of the series
ln x2 + ln¼
giving your answer in the form ln, where m, n Î .
(Total 5 marks)
7. The function f is defined for x > 2 by f (x) = ln x + ln (x – 2) – ln (x2 – 4).
(a) Express f (x) in the form ln.
(b) Find an expression for f –1(x).
(Total 6 marks)
8. Let y = log3 z, where z is a function of x. The diagram shows the straight line L, which represents the graph of y against x.
(a) Using the graph or otherwise, estimate the value of x when z = 9.
(b) The line L passes through the point . Its gradient is 2. Find an expression for z in terms of x.
(Total 6 marks)