1. Exponentials and Logarithms
Objectives
Know and use the functions 𝑎 𝑥 and 𝑒 𝑥 and their graphs, where 𝑎 is positive.
Know and use the definition of log 𝑎 𝑥 as the inverse of 𝑎 𝑥 , where 𝑎 is positive and 𝑥 ≥ 0.
Know and use ln 𝑥 as the inverse function of 𝑒 𝑥 .
Know and use the function ln 𝑥 and its graph.
Understand and use the laws of logarithms:
log 𝑎 𝑥 + log 𝑎 𝑦 =
log 𝑎 𝑥𝑦
log 𝑎 𝑥 𝑦
log 𝑎 𝑥 − log 𝑎 𝑦 =
log 𝑎 𝑥 𝑛 =
𝑛 log 𝑎 𝑥
Solve equations of the form 𝑎 𝑥 =𝑏.

2. Exponentials and Logarithms
Quick Links
Introducing Natural Logarithms and ‘e’

3. Graphs of Exponentials
Use Desmos to sketch the curves below on the same axis.
Give the coordinates of any points where the graphs cross the axis.
(i)
𝑦= 2 𝑥
(ii)
𝑦=3 𝑥
(iii)
𝑦= 0.5 𝑥
𝑦= 3 𝑥
𝑦= 2 𝑥
𝑦= 0.5 𝑥
The circles at the end of the domain are not required knowledge for A-Level
Note:
𝑦= 2 𝑥 is steeper than 𝑦= 3 𝑥
𝑦= 0.5 𝑥 is a reflection of 𝑦= 2 𝑥 in the 𝑦-axis.

4. Exponentials, Roots and Logarithms
Exponential question: 3 2 = ?
Root question: ? 2 =9
This can be written another way:
log 3 9 =?
“Log base 3 of 9”
Logarithm question: 3 ? =9
(The answer is ‘2’)
A logarithm answers the question
“How many of one number do we multiply to get another number?”
We use logs when we want to find an unknown power.
The circles at the end of the domain are not required knowledge for A-Level
Logs are the inverse of exponentials. More specifically,
log 𝑎 𝑥 is the inverse of 𝑎 𝑥 (where 𝑎 is positive and 𝑥≥0)

5. Introduction to Logarithms
You have previously met logarithm questions, although this may not be what you called the. For example:
1 4
1 2
(i)
3 𝑥 = 27
(ii)
81 𝑥 = 9
(iii)
4 𝑥 =
(iv)
32 𝑥 =
𝑥=3
𝑥= 1 2
𝑥=−1
𝑥= − 1 5
However, not all logarithm questions are so straightforward.
Try to solve the logarithm equation below using trial and improvement.
Give your answer to one decimal place.
3 𝑥 =7
1.8 is closest

6. Exponentials and Logarithms
Terminology
exponent
“is equivalent to”
Tip: Notice how the ‘base’ is the same in both forms.
4 3 =64 ⇔ log =3
base
base
argument
Exercise
Write the following in the equivalent form:
log =−3
⇔ 2 −3 = 1 8
(i)
5 3 =125
⇔ log =3
(ii)
log 36 6= 1 2
⇔ =6
81 − 1 4 = 1 3
⇔ log =− 1 4
(iv)
(iii)

7. Special Cases Base 10 Logarithms log 10 𝑥 = log 𝑥
If no base is stated, we assume it is base 10.
Same Base
The following applies to logarithms of any base:
If the base and the argument are the same, the log evaluates to 1.
log 𝑎 𝑎 =
e.g. log 3 3 = 1 … 1
(because 𝑎 1 =𝑎)
If the argument is 1, the log evaluates to 0 …
log 𝑎 1 =
(because 𝑎 0 =1)

8. Evaluating Logs | Example-Problem Pairs
Copy down each example in part (a) then try part (b) for yourself.
Example 1
2 𝑥 =8
Evaluate
(a)
log 2 8= 3
(b)
log 2 16= 4
16 𝑥 =4
Example 2
log =
1 2
− 1 3
(a)
log 16 4=
(b)
Next…
C2 Solomon Press – Worksheet A

9. Evaluating Logarithms - Exercise
C2 Solomon Press A
Complete the following questions from the C2 Solomon Press A worksheet:
Q1: b, d, f
Q2: a, e, g
Q3: All parts
Q4: All parts

10. The Logarithmic Identities (Log Laws)
The following logarithmic identities are always applicable:
(i)
log 𝑎 𝑥 + log 𝑎 𝑦 ≡
log 𝑎 𝑥𝑦
(addition law)
log 𝑎 𝑥 𝑦
(ii)
log 𝑎 𝑥 − log 𝑎 𝑦 ≡
(subtraction law)
(iii)
log 𝑎 𝑥 𝑛 ≡
𝑛 log 𝑎 𝑥
(power law)
It is important to memorise these laws as soon as possible.

11. The Logarithmic Identities (Log Laws)
Example 1
6×4=24
log 𝑎 𝑥 + log 𝑎 𝑦 ≡
log 𝑎 𝑥𝑦
log 𝑎 𝑥 𝑦
log log 3 4 =
log 3 24
log 𝑎 𝑥 − log 𝑎 𝑦 ≡
Example 2
24÷3=8
log 𝑎 𝑥 𝑛 ≡
𝑛 log 𝑎 𝑥
log − log 4 3 =
log 4 8
Notice that the base must be the same for both terms when using the addition and subtraction law.
Example 3
“bring the power down”
log =
2log 5 13

12. Simplifying Logarithmic Expressions
On your whiteboards, express the following in the form log 𝑎 𝑛 .
(a)
log log 3 8 =
log 3 40
(b)
2log log 2 3 =
log log 2 3
=log
1 2 log 16 − 1 5 log 32 =
(c)
log 4 − log 2
= log 2
Tarzia Puzzle
The circles at the end of the domain are not required knowledge for A-Level
Work in pairs to complete the simplifying logs tarsia puzzle.
C2 Solomon Press Worksheet A
Complete the following questions from the C2 Solomon Press A worksheet:
Q8: All parts

13. Solving Logarithmic Equations
There are two standard methods for solving logarithmic equations.
Example 1
Solve
2 𝑥 =5
Give your answer to two decimal places.
Method 1 – Equivalent Form
Method 2 – Any Base
𝑥=2.32

14. Using Logs to Solve Equations
There are two standard methods for solving logarithmic equations.
Example 2
Solve
3 2𝑥−5 =12
Give your answer to two decimal places.
Method 1 – Equivalent Form
Method 2 – Any Base
𝑥=3.63

15. Solving Exponential Equations | Exercise
Solve the following equations. Give your answers to 3 significant figures.
2× 𝑥 −0.6=0
(a)
3 𝑥 =12
(b)
𝑥=2.26
𝑥=−1.74
(c)
7 2𝑥+4 =12
(d)
5(2 3𝑥+1 )=62
𝑥=−1.36
𝑥=0.877
(e)
4 5−𝑥 = 11 2𝑥−1
(f)
3 2𝑥 = 3 𝑥−1 × 2 4+𝑥
The circles at the end of the domain are not required knowledge for A-Level
𝑥=1.51
𝑥=4.13
Tip for part (f): You can simplify using ‘power’ laws

16. Solving Equations Containing Logs
Copy down the example in part (a) then try part (b) for yourself.
Example 1
Solve the following equations. Leave your answer as an exact value.
1 2 log 5𝑥 +3=8
(a)
(b)
3 log 𝑥 −1=5
A single term contains a log.
𝑥=100
𝑥=2× 10 9

17. Solving Equations Containing Logs
Copy down the example in part (a) then try part (b) for yourself.
Example 2
Solve the following equations. Leave your answer as an exact value.
(a)
(b)
log 6𝑥 + log 2 = log (5𝑥+4)
2 log 𝑥 + log 4 = log (2𝑥+1)
Every term contains a log.
𝑥= 4 7
1± 5 4

18. Solving Equations Containing Logs
Copy down the example in part (a) then try part (b) for yourself.
Example 3
Solve the following equations. Leave your answer as an exact value.
log 𝑥 − log 3 = 3 4
3 log 4𝑥 = log 8𝑥 + 2 5
(a)
(b)
Some terms contain logs. 3×
1 2 ×

19. Solving Equations Containing Logs | Exercise
Solomon Press B
Complete the following questions from the Solomon Press B worksheet:
Q7: All parts

20. Exponentials and Logarithms
Exponential Growth
Exponential growth occurs where the rate of growth of a function is proportional to its current value.
A Special Base – Euler’s Number
Link to Desmos Activity: goo.gl/hyq2UV
Use Desmos to view the graphs of 𝑓(𝑥)= 2 𝑥 and its derivative.
Notice that 𝑓 𝑥 > 𝑓 ′ 𝑥 for all values of 𝑥.
Now view the graphs of 𝑔(𝑥)= 3 𝑥 and its derivative.
Notice that 𝑔 𝑥 < 𝑔 ′ 𝑥 for all values of 𝑥.
Use the slider the find a value such that ℎ 𝑥 = ℎ ′ 𝑥 (to 2 d.p.)

21. Natural Logarithms and ‘e’
Euler’s Number
𝑒 = … (𝑒 is an irrational number)
Natural Logarithms
Previously, you learned how to convert between exponential and logarithmic forms.
Examples
(i)
log 2 8 =3
(ii)
log 2 𝑦 =𝑥
2 3 =8 ⇔
𝑦= 2 𝑥 ⇔
Therefore,
𝑦= 𝑒 𝑥 ⇔
log 𝑒 𝑦 =𝑥
However, we generally write log 𝑒 𝑦 as ln 𝑦 (ln stands for ‘natural logarithm’)
Note:
𝑒 𝑥 𝑎𝑛𝑑 ln 𝑥 are inverse functions

22. Solving Equations Involving ‘𝑒’
Solve the following equations. Leave your answer as an exact value.
Example 1
Example 2
log 𝑒 𝑒 = 1
1 6 𝑒 4𝑥+3 −2=7
(a)
𝑒 𝑥 =8
i.e.
ln 𝑒=1
(b)

23. Solving Equations Involving ‘e’| Exercise
C3 Solomon Press A
Complete the following questions from the Solomon Press B worksheet:
Q5: All parts

24. Logarithms and Hidden Quadratics
Solve the following equations. Leave your answer as an exact value.
Example 1
Example 2
(a)
ln 6𝑥−2 =4
(b)
ln 4𝑥+6 − ln 𝑥 =ln⁡(3𝑥+1)
1 6 (𝑒 4 +2)
Changes: Split this into two separate examples with questions for each.

25. Exponentials and Logarithms | Exercise
1. Solve the following equations.
(i) ln 𝑥 + ln 5= ln (2𝑥+3)
(ii) ln (3𝑥−4) − ln (𝑥−3)= ln 2𝑥
(iii) ln (5𝑥) − ln (𝑥+2)= ln 𝑥+6 − ln 𝑥
(iv) 2 ln 𝑥 = ln 2𝑥−5 + ln 5
Answers
(i)
𝑥=1
(ii)
𝑥=4
(iii)
𝑥=3,
(iv)
𝑥=5
2. Solve the following equations. Give your answers to 3 significant figures.
ln 4𝑥 =ln 1 𝑥−6 +1
ln 𝑥 −ln 𝑥−1 = ln
(i)
(ii)
(iii)
Answers
(i)
𝑥=6.11 (3 s.f.)
(ii)
𝑥=1.25

26. Logarithms and Hidden Quadratics’
Solve the following equations. Leave your answer as an exact value.
Example 1
Example 2
(a)
2𝑒 2𝑥 +12=11 𝑒 𝑥
(b)
2 𝑒 𝑥 +3 𝑒 −𝑥 =7
𝑥=ln , ln 4
ln , ln 3

27. Logarithms and Hidden Quadratics | Exercise
Solving exponential equations by reducing to quadratic form.
(i)
𝑒 2𝑥 −12 𝑒 𝑥 +27=0
(give your answer as an exact value)
(ii)
𝑒 2𝑥 +4 𝑒 𝑥 −5=0
(iii)
𝑒 2𝑥 −5.7 𝑒 −𝑥 =0
(give your answer as an exact value)
(iv)
5𝑒 𝑥 +15 𝑒 −𝑥 =17
(give your answer as an exact value)
(v)
𝑒 2𝑥+1 −26 𝑒 𝑥 +5=0
(give your answer to 3 s.f.)
Answers
𝑥= 1 3 ln 5.7
(i)
𝑥= ln 9 , ln 3
(ii)
𝑥=0
(iii)
𝑥= ln 3 , ln 5 4
(iv)
(v)
𝑥=2.24, −1.63