Worked examples for the Maths IA topic 'Logarithms and Exponentials'

Mathematics IA
Worked Examples
CALCULUS: THE LOGARITHM AND
EXPONENTIAL FUNCTIONS
Produced by the Maths Learning Centre,
The University of Adelaide.
May 3, 2013
The questions on this page have worked solutions and links to videos on
the following pages. Click on the link with each question to go straight to
the relevant page.
Questions
1. See Page 3 for worked solutions.
(a) Let a > 0. Find the derivative of the function ln(ax).
(b) Using the definition of the function ln x and the Fundamental Theorem
of Calculus, show that ln a + ln b = ln(ab), for all a, b > 0.
2. See Page 4 for worked solutions.
2.5
1
Use the upper sum U6 for dt to show that e > 2.5.
t
3. See Page 6 for worked solutions.
Using the properties of ln x, show that (e a ) b = e ab .
4. See Page 7 for worked solutions.
Solve the following equations for x ∈ R:
(a) ln(x + 3) + ln(x − 3) = 3 (b) 3e x = 2 + 5e −x .
5. See Page 9 for worked solutions.
Find the derivatives of the following functions:
(a) ln(5x 4 ) (b) x 2 e x
6. See Page 10 for worked solutions.
Find the derivatives of the following functions:
(a) log10(x + 2) (b) 3 · 23x ln x (c) x
1
1

7. See Page 12 for worked solutions.
Use the definitions of sinh x and cosh x to show that
cosh(2x) − 1 .
sinh 2 x = 1
2
8. See Page 13 for worked solutions.
Let arctanh x be the inverse function of tanh x. Write down the domain
and range of arctanh x and find a formula for arctanh x.
9. See Page 15 for worked solutions.
Find the derivatives of the following functions:
(a) sinh(x 2 ) (b) tanh 3 (x) (c) cosh x cos x
2

1. Click here to go to question list.
(a) Let a > 0. Find the derivative of the function ln(ax).
(b) Using the definition of the function ln x and the Fundamental Theorem
of Calculus, show that ln a + ln b = ln(ab), for all a, b > 0.
Link to video on YouTube
3

2. Click here to go to question
list.
2.5
1
Use the upper sum U6 for dt to show that e > 2.5.
1 t
Link to video on YouTube
4

3. Click here to go to question list.
Using the properties of ln x, show that (e a ) b = e ab .
Link to video on YouTube
6

4. Click here to go to question list.
Solve the following equations for x ∈ R:
(a) ln(x + 3) + ln(x − 3) = 3 (b) 3e x = 2 + 5e −x .
Link to video on YouTube
7

5. Click here to go to question list.
Find the derivatives of the following functions:
(a) ln(5x 4 ) (b) x 2 e x
Link to video on YouTube
9

6. Click here to go to question list.
Find the derivatives of the following functions:
(a) log10(x + 2) (b) 3 · 23x ln x (c) x
Link to video on YouTube
10

7. Click here to go to question list.
Use the definitions of sinh x and cosh x to show that
cosh(2x) − 1 .
sinh 2 x = 1
2
Link to video on YouTube
12

8. Click here to go to question list.
Let arctanh x be the inverse function of tanh x. Write down the domain
and range of arctanh x and find a formula for arctanh x.
Link to video on YouTube
13

9. Click here to go to question list.
Find the derivatives of the following functions:
(a) sinh(x 2 ) (b) tanh 3 (x) (c) cosh x cos x
Link to video on YouTube
15