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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)

 

 

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