Let be an exponential function satisfying the following conditions: • The point lies…

MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms

Problem

Let

f(x)=a\,b^{x}

be an exponential function satisfying the following conditions: • The point

(0,a)

lies on the graph. • The point

(2,45)

lies on the graph. • Both

a

and

b

are positive integers, with

b>1

and

1\le a\le 10

. (i) Show that

b^{2}

must be a factor of

45

and hence determine the possible integer value of

b

. (ii) For the value of

b

obtained, find the corresponding value of

a

. (iii) Using the values of

a

and

b

, calculate

f(4)