Exponential and Logarithmic Functions – Extra Practice
1.
Let f(x) = log3
x
+ log3 16 – log3 4, for x > 0.
2
(a)
Show that f(x) = log3 2x.
(b)
Find the value of f(0.5) and of f(4.5).
The function f can also be written in the form f(x) =
(c)
(d)
ln ax
.
ln b
(i)
Write down the value of a and of b.
(ii)
Hence on graph paper, sketch the graph of f, for –5 ≤ x ≤ 5, –5 ≤ y ≤ 5, using a
scale of 1 cm to 1 unit on each axis.
(iii)
Write down the equation of the asymptote.
Write down the value of f–1(0).
The point A lies on the graph of f. At A, x = 4.5.
(e)
2.
On your diagram, sketch the graph of f–1, noting clearly the image of point A.
Let f(x) = log3 x , for x > 0.
(a)
Show that f–1(x) = 32x.
(b)
Write down the range of f–1.
Let g(x) = log3 x, for x > 0.
(c)
3.
Find the value of (f –1 ° g)(2), giving your answer as an integer.
Let f(x) = ex+3.
(a)
(b)
(i)
Show that f–1(x) = ln x – 3.
(ii)
Write down the domain of f–1.
1
Solve the equation f–1(x) = ln  .
 x
4.
Let f (x) = ln (x + 5) + ln 2, for x > –5.
(a)
Find f −1(x).
Let g (x) = ex.
(b)
5.
Find (g ◦ f) (x), giving your answer in the form ax + b, where a, b,∈
.
The mass m kg of a radio-active substance at time t hours is given by
m = 4e–0.2t.
6.
(a)
Write down the initial mass.
(b)
The mass is reduced to 1.5 kg. How long does this take?
A group of ten leopards is introduced into a game park. After t years the number of leopards,
N, is modelled by N = 10 e0.4t.
(a)
How many leopards are there after 2 years?
(b)
How long will it take for the number of leopards to reach 100? Give your answers to
an appropriate degree of accuracy.
Give your answers to an appropriate degree of accuracy.