Logarithmic ppt

1. LOGARITHMS
Functions, Equations, Inequalities
1

2. Definition: Let a and b be
positive numbers such that 𝑏 ≠
1. The logarithm of a with the
base b denoted by
𝑙𝑜𝑔𝑏𝑎
and is defined as
𝑐 = 𝑙𝑜𝑔𝑏𝑎 iff 𝑎 = 𝑏𝑐
REMINDERS:
1. logarithmic and
exponential functions
are inverses.
2. In 𝑙𝑜𝑔𝑏𝑥 = 𝑥, x
can’t be negative.
3. the value of 𝑙𝑜𝑔𝑏𝑥
can be negative.

3. COMMON
LOGARITHMS
Definition:
Common Logarithms
are logarithms with the
base 10;
𝑙𝑜𝑔𝑥 is a short
notation for 𝑙𝑜𝑔10𝑥.

4. NATURAL
LOGARITHMS
Definition:
Natural Logarithms
are logarithms to the
base 𝑒 (2.71828)
denoted by “ln”.
𝑙𝑛𝑥 is another way of
writing 𝑙𝑜𝑔𝑒𝑥

5. 53
= 125
40
= 1
7−2
=
1
49
102
= 100
𝑙𝑜𝑔5125 = 3
𝑙𝑜𝑔41 = 0
𝑙𝑜𝑔7
1
49
= −2
𝑙𝑜𝑔100 = 2
REWRITE THE EXPONENTIAL FUCTIONS
TO LOGARITHMIC FUNCTIONS

6. 10𝑛
= 𝑚
34
= 81
5
2
= 5
41/2
= 2
3
4
−3
=
64
27
log m = n
log3 81 = 4
log 5 5 = 2
log4 2 = 1/2
log3
4
64
27
= −3
REWRITE THE LOGARITHMIC FUNCTIONS
TO EXPONENTIAL FUCTIONS

7. 1. TRANSFORM LOGARITHMIC EXPRESSIONS
TO EXPONENTIAL EXPRESSIONS
2. SOLVE FOR THE UNKNOWN.

8. Definition:
An equation involving
logarithms.
Example:
𝑙𝑜𝑔𝑥2 = 4
LOGARITHMIC
EQUATION

9. Definition:
An inequality involving
logarithms.
Example:
ln 𝑥2
> (ln 𝑥)2
LOGARITHMIC
INEQUALITY

10. Definition:
A function in the form
𝒇 𝒙 = 𝒍𝒐𝒈𝒃 𝒙
(𝑏 > 0; 𝑏 ≠ 1)
Example:
𝒈 𝒙 = 𝒍𝒐𝒈𝟑 𝒙
LOGARITHMIC
FUNCTIONS

11. LOGARITHMS
BASIC PROPERTIES
11

12. Definition:
et a and x be real numbers
such that
𝑏 > 0 , 𝑏 ≠ 1. The basic
properties of logarithms
are:
1. 𝑙𝑜𝑔𝑏1 = 0
2. 𝑙𝑜𝑔𝑏𝑏𝑥
= 𝑥
3. If 𝑥 > 0, then
𝑏𝑙𝑜𝑔𝑏𝑥
= 𝑥

13. 1.log10
2.lne3
3.log464
Use the properties logb1=0, 𝑙𝑜𝑔𝑏𝑏𝑥
= 𝑥 , and 𝑏𝑙𝑜𝑔𝑏𝑥
= 𝑥 to
find the value of the following logarithmic expressions.
1 (property 2)
3 (property 2)
3 (property 2)

14. 4. log5(1/125)
5. 5log5 2
6. log1
Use the properties logb1=0, 𝑙𝑜𝑔𝑏𝑏𝑥
= 𝑥 , and 𝑏𝑙𝑜𝑔𝑏𝑥
= 𝑥 to
find the value of the following logarithmic expressions.
-3 (property 2)
2 (property 3)
0 (property 1)

15. FIND THE VALUE USING THE BASIC
PROPERTIES OF LOGARITHMS.
1. log99
2. log4(1/16)
3. 10log10 5
4. lne4
5. log41
1. 1 (prop. 2)
2. - 2 (prop. 2)
3. 5 (prop. 3)
4. 4 (prop. 2)
5. 0 (prop. 1)

16. FIND THE VALUE USING THE BASIC
PROPERTIES OF LOGARITHMS.
1. 𝑙𝑜𝑔5 5
2. 𝑙𝑜𝑔11
1
121
3. 𝑒𝑙𝑛 6
4. 𝑙𝑛 𝑒𝑥+1
5. 𝑙𝑜𝑔9 1
1. 1 (prop. 2)
2. -2 (prop. 2)
3. 6 (prop. 3)
4. x+1 (prop. 2)
5. 0 (prop. 1)

17. LAWS OF
LOGARITHMS
17

18. 1.log𝑏(𝑢𝑣) = log𝑏 𝑢 + log𝑏 𝑣
2.log𝑏(𝑢/𝑣) = log𝑏 𝑢 − log𝑏 𝑣
3.log𝑏 𝑢𝑛
= 𝑛 ∙ log𝑏 𝑢
Let b > 0, b ≠ 1 and let n ϵ R.
For u > 0, v > 0
• log2(3𝑥) = log2 3 + log2 𝑥
• log3(4/5) = log3 4 − log3 5
• log5 36 = log5 62
= 2 log5 6

19. • log2(5 + 2) ≠ log2 5 + log2 2
• log2(5 + 2) ≠ (log2 5) (log2 2)
• log2(5 — 2) ≠ log2 5 — log2 2
• log2(5 — 2) ≠
log2 5
log2 2
• log2(52 • 2) ≠ 2 log2(5 • 2)
COMMON MISTAKES

20. •log 2 + log 3
•2ln x — In y
•log5(x2) — 3 log5 x
•2 — log 5
Use the
properties of
logarithm to
condense the
expressions
as a single
logarithm.
•log 6
•ln (𝑥2
/ y)
•−log5 𝑥
•log 20

21. •log 3 — log 8
•ln x — 4In y
•log2 + log3 + log4
•
log6
3
Use the
properties of
logarithm to
condense the
expressions
as a single
logarithm.
•log 3/8
•ln (𝑥/y4
)
•𝑙𝑜𝑔 24
•log
3
6

22. 1. log 7 + 2log 3
2. 2log310 — log34
3. ln2 + lnx — ln5
Q3: Use the
properties of
logarithm to
condense the
expressions
as a single
logarithm.
log63
log325
ln
2x
5

23. 4.
2𝑙𝑜𝑔2
𝑥
3
5. log4y + 3
Q3: Use the
properties of
logarithm to
condense the
expressions
as a single
logarithm.
log
3
𝑥2
log4(64y)