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Chapter 6 Logarithmic and exponen�al func�ons
6.1 Logarithms to base 10
6.2 Logarithms to base a
6.3 The laws of logarithms
6.4 Solving logarithmic equa�ons
6.5 Solving exponen�al equa�ons
6.6 Change of base of logarithms
6.7 Natural logarithms
6.8 The inverse of logarithmic and exponen�al func�ons
Logarithmic Defini�on
Exponen�al form
Logarithmic form
if 𝒚𝒚 = 𝒂𝒂𝒙𝒙 ,
then 𝒙𝒙 = 𝐥𝐥𝐥𝐥𝐥𝐥 𝒂𝒂 𝒚𝒚
The laws of logarithmic
Laws for logarithms
Opera�on
log 𝑎𝑎 𝑥𝑥𝑥𝑥 = log 𝑎𝑎 𝑥𝑥 + log 𝑎𝑎 𝑦𝑦
Mul�plica�on
log 𝑎𝑎 𝑥𝑥 𝑛𝑛 = 𝑛𝑛 log 𝑎𝑎 𝑥𝑥
Powers
log 𝑎𝑎
𝑥𝑥
= log 𝑎𝑎 𝑥𝑥 − log 𝑎𝑎 𝑦𝑦
𝑦𝑦
1
= − log 𝑎𝑎 𝑥𝑥
𝑥𝑥
1
𝑛𝑛
log 𝑎𝑎 √𝑥𝑥 = log 𝑎𝑎 𝑥𝑥
𝑛𝑛
log 𝑐𝑐 𝑏𝑏
log 𝑎𝑎 𝑏𝑏 =
log 𝑐𝑐 𝑎𝑎
Reciprocals
log 𝑎𝑎
log 𝑎𝑎 𝑏𝑏 =
Division
Roots
Change of base
1
log 𝑏𝑏 𝑎𝑎
Change of base
Iden�ty for logarithmic
log 𝑎𝑎 1 = 0
log 𝑎𝑎 𝑏𝑏 log 𝑏𝑏 𝑐𝑐 = log 𝑎𝑎 𝑐𝑐
log 𝑎𝑎 𝑎𝑎𝑟𝑟 = 𝑟𝑟
𝑥𝑥 = 𝑎𝑎log𝑎𝑎 𝑥𝑥
log 𝑎𝑎 𝑎𝑎 = 1
log 1 𝑏𝑏 = − log 𝑎𝑎 𝑏𝑏
𝑎𝑎
log 𝑎𝑎𝑚𝑚 𝑏𝑏 𝑛𝑛 =
n
𝑚𝑚
𝑚𝑚 ≠ 0
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Chapter 6 Logarithmic and exponen�al func�ons
6.1 Logarithms to base 10
1. Convert from exponen�al form to logarithmic form and solve it.
a) 10𝑥𝑥 = 30
b) 10𝑥𝑥 = 0.06
c) 10𝑥𝑥 = 720
2. Convert from logarithmic form to exponen�al form and solve it.
a) log10 𝑥𝑥 = 2.9
b) log10 𝑥𝑥 =
1
2
c) log10 𝑥𝑥 = −0.8
3. Find the value of the following using Logarithmic Rules.
a) log10 100
b) log10 0.001
c) log10 100 √10
Addmath
Chapter 6 Logarithmic and exponen�al func�ons
6.2 Logarithms to base a
1. Convert from exponen�al form to logarithmic form
a) 24 = 16
b) 2−5 =
1
32
c) 𝑥𝑥 𝑦𝑦 = 4
2. Convert from logarithmic form to exponen�al form
a) log 7 49 = 2
b) log 𝑥𝑥 8 = 𝑦𝑦
c) log 𝑎𝑎 𝑏𝑏 = 𝑐𝑐
3. Find the value of the following using Logarithmic Rules.
a) log 3 81
b) log 2 (8√2)
1
c) log 2 � �
√8
4. Solve the following using Logarithmic Rules.
a) log 2 𝑥𝑥 = 4
b) log 3 (2𝑥𝑥 + 1) = 2
c) log 𝑥𝑥 144 = 2
5. Simplify the following using Logarithmic Rules.
a) log 𝑥𝑥 𝑥𝑥 2
b) log 𝑥𝑥 (𝑥𝑥 √𝑥𝑥)
c) log 𝑥𝑥
(𝑥𝑥√𝑥𝑥)
3
√𝑥𝑥
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Chapter 6 Logarithmic and exponen�al func�ons
6.3 The laws of logarithms
1. Use the laws of logarithms to simplify these expressions.
a) lg 8 + lg 2
b) log 4 15 ÷ log 4 5
c) 2 log 3 4 + 5 log 3 2
2. Given that log 5 𝑝𝑝 = 𝑥𝑥 and log 5 𝑞𝑞 = 𝑦𝑦, express in terms of 𝑥𝑥 and/or 𝑦𝑦.
a) log 5 𝑝𝑝 + log 5 𝑞𝑞 3
3. Given that 𝑢𝑢 = log 5 𝑥𝑥, find, in simplest form in term of 𝑢𝑢.
a) 𝑥𝑥
𝑥𝑥
b) log 5 � �
25
4. Given that log 𝑎𝑎 𝑥𝑥 = 12 and log 𝑎𝑎 𝑦𝑦 = 4, find the value of
𝑥𝑥
a) log 𝑎𝑎 � �
𝑦𝑦
𝑞𝑞
b) log 5 𝑝𝑝2 − log 5 �𝑞𝑞
𝑥𝑥 2
b) log 𝑥𝑥 � �
𝑦𝑦
c) log 5 �
𝑥𝑥√𝑥𝑥
125
c) log 5 � �
5
�
c) log 𝑎𝑎 �𝑥𝑥 �𝑦𝑦�
d) log 5 �5√𝑥𝑥�
𝑦𝑦
d) log 𝑎𝑎 � 3 �
√𝑥𝑥
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Chapter 6 Logarithmic and exponen�al func�ons
6.4 Solving logarithmic equa�ons
a) 4 log 𝑥𝑥 2 − log 𝑥𝑥 4 = 2
b) 2 log 8 (𝑥𝑥 + 2) = log 8 (2𝑥𝑥 + 19)
c) lg(4𝑥𝑥 + 5) + 2lg2 = 1 + lg (2x − 1)
d) (log 5 𝑥𝑥 )2 − 3log 5 𝑥𝑥 + 2 = 0
e) Solve the logarithmic simultaneous equa�ons (Challenge)
𝑥𝑥𝑥𝑥 = 64
log 𝑥𝑥 𝑦𝑦 = 2
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Chapter 6 Logarithmic and exponen�al func�ons
6.5 Solving exponen�al equa�ons (Famous ques�on)
a) 3𝑥𝑥 = 40
b) 52𝑥𝑥+1 = 200
d) Solve the exponen�al quadra�c equa�ons (Challenge)
i.
32𝑥𝑥 − 6 × 3𝑥𝑥 + 5 = 0
ii.
42𝑥𝑥 − 6 × 4𝑥𝑥 − 7 = 0
c) 32𝑥𝑥+3 = 53𝑥𝑥+1
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Chapter 6 Logarithmic and exponen�al func�ons
6.6 Change of base of logarithms
1. Change log 2 7 to base 10. Hence evaluate log 2 7 correct to 3 sf.
2. Solve log 3 𝑥𝑥 = log 9 (𝑥𝑥 + 6).
3. Given that u = log 4 𝑥𝑥, find, in simplest form in terms of u.
a) log 𝑥𝑥 4
b) log 𝑥𝑥 16
c) log 𝑥𝑥 2
d) log 𝑥𝑥 8
4. Given that log 𝑝𝑝 𝑥𝑥 = 20 and log 𝑝𝑝 𝑦𝑦 = 5, find log 𝑦𝑦 𝑥𝑥
5. Express log 4 𝑥𝑥 in terms of log 2 𝑥𝑥. Using your answer and subs�tuion 𝑢𝑢 = log 2 𝑥𝑥, solve the
equa�on log 4 𝑥𝑥 + log 2 𝑥𝑥 = 12.
6. Solve
a) log 2 𝑥𝑥 + 5 log 4 𝑥𝑥 = 14
b) 5 log 2 𝑥𝑥 − log 4 𝑥𝑥 = 3
7. Solve
a) log 3 𝑥𝑥 = 9 log 𝑥𝑥 3
b) log 5 𝑥𝑥 + log 𝑥𝑥 5 = 2
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Chapter 6 Logarithmic and exponen�al func�ons
6.7 Natural logarithms
What is natural logarithmic?
It is anotehr type of logarithmic to the base of e. The number e = 2.718. Logarithms to the base
of e are called natural logarithms.
Natural logarithmic defini�on
Exponen�al form
Logarithmic form
if 𝒚𝒚 = 𝒆𝒆𝒙𝒙 ,
then 𝒙𝒙 = log 𝑒𝑒 𝑦𝑦 = 𝐥𝐥𝐥𝐥 𝒚𝒚 (ln = log 𝑒𝑒 )
1. Without using a calculator find the value of
a) eln 5
1
b) e2 ln 64
c) 3eln 2
b) ln ex = 2.5
c) e2 ln x = 36
2. Solve.
a) eln x = 7
3. Solve, giving your ansers in terms of natural logarithms.
a) ex = 7
b) 2ex + 1 = 7
c) e2x−5 = 3
4. Solve, giving your answers correct to 3 sf.
a) ln 𝑥𝑥 3 + ln 𝑥𝑥 = 5
b) e3x+4 = 2ex−1
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Chapter 6 Logarithmic and exponen�al func�ons
6.8 The inverse of logarithmic and exponen�al func�ons
1. Find the inverse of each func�on and state its domain.
b) 𝑓𝑓(𝑥𝑥 ) = 3ln (2𝑥𝑥 − 4) for 𝑥𝑥 > 2
a) 𝑓𝑓(𝑥𝑥 ) = 2𝑒𝑒 −4𝑥𝑥 + 3
for 𝑥𝑥 ∈ 𝑅𝑅
2. Find 𝑓𝑓 −1 (𝑥𝑥 ) for each func�on and state its domain.
a) 𝑓𝑓(𝑥𝑥 ) = 𝑒𝑒 𝑥𝑥 + 4
b) 𝑓𝑓(𝑥𝑥 ) = 3𝑒𝑒 2𝑥𝑥 + 1
3. Find 𝑓𝑓 −1 (𝑥𝑥 ) for each func�on.
a) 𝑓𝑓(𝑥𝑥 ) = ln(𝑥𝑥 + 1), 𝑥𝑥 > −1
b) 𝑓𝑓(𝑥𝑥 ) = 2 ln(𝑥𝑥 + 2), 𝑥𝑥 > −2
4. 𝑓𝑓(𝑥𝑥 ) = 𝑒𝑒 𝑥𝑥 for 𝑥𝑥 ∈ 𝑅𝑅
𝑔𝑔(𝑥𝑥 ) = ln 5𝑥𝑥 for 𝑥𝑥 > 0
a) Find 𝑓𝑓𝑔𝑔(𝑥𝑥 ) & 𝑔𝑔𝑔𝑔 (𝑥𝑥 ).
b) Solve 𝑔𝑔(𝑥𝑥 ) = 3𝑓𝑓 −1 (𝑥𝑥 ).
𝑔𝑔(𝑥𝑥 ) = ln 𝑥𝑥 for 𝑥𝑥 > 0
5. 𝑓𝑓(𝑥𝑥 ) = 𝑒𝑒 3𝑥𝑥 for 𝑥𝑥 ∈ 𝑅𝑅
c) Find 𝑓𝑓𝑔𝑔(𝑥𝑥 ) & 𝑔𝑔𝑔𝑔 (𝑥𝑥 ).
d) Solve 𝑓𝑓(𝑥𝑥 ) = 2𝑔𝑔−1 (𝑥𝑥 ).