Logarithm Rules

Before doing this exercise, you should know about the Definition of the Logarithm.

When a>0,a≠1,M>0
logaM=x means ax=M

Logarithm Rules
When a>0. a≠1,M>0, N>0

1.logaM+logaN=logaMN

2.logaM−logaN=loga

3.n logaM=loga(M n) Simply written =logaM n

Logarithm of Special Arguments
For any base a>0, a≠1,

Iloga1=0
IIlogaa=1

(Proof)
Suppose ax=M, ay=N,
then x=logaM, y=logaN ←Definition of Logarithm
1.←

ax+y=axay	→	ax+y=MN	Exponential form
		↓	--Convert--
logaM+logaN=logaMN	←	x+y=logaMN	Logarithmic form

2.←

ax−y=	→	ax−y=	Exponential form
		↓	--Convert--
logaM−logaN=loga	←	x−y=loga	Logarithmic form

3.←

(ax) n=M n	→	a nx=M n	Exponential form
		↓	--Convert--
n logaM=logaM n	←	nx=logaM n	Logarithmic form

I←
a0=1→loga1=0
II←
a1=a→logaa=1

Examples

log28=3 means 23=8
log10100=2 means 102=100

Examples
1.
log102+log103=log102×3=log106

log54+log512=log54×12=log548
2.
log210−log25=log2 =log22

log618−log62=log6 =log69
3.
2 log310=log3(10 2)=log3100

3 log22=log2(2 3)=log28
I
log21=0

log101=0
II
log22=1

log33=1

# Common Mistakes #

-- Incorrect --	-- Correct --
loga(M+N)≠logaMN Not the logarithm of a sum	logaM+logaN=logaMN But the sum of the logarithms
logaM·logaN≠logaMN Not the product of the logarithms	logaM+logaN=logaMN But the sum of the logarithms
logaM+logaN≠logaM·logaN Not the product of the logarithms	logaM+logaN=logaMN But the product of the arguments
loga(M−N)≠loga Not the difference of the arguments	logaM−logaN=loga But the difference of the logarithms
≠loga Not the quotient of the logarithms	logaM−logaN=loga But the difference of the logarithms
logaM−logaN≠ Not the quotient of the logarithms	logaM−logaN=loga But the quotient of the arguments
n logaM≠(loga M) n Not the power of the logarithm	n logaM=loga(M n) But the power of the argument

In many cases, in order to simplify the logarithmic expressions, you had better express as a single logarithm.

e.g.
1.log102+log105 → log1010

2.log1050−log105 → log10

Thus, in order to simplify the logarithmic expressions, the above Logarithm Rules work fine when converted the left side into the right side.

Example 1Simplify each of the following logarithms.

(1)log52+log53

(Answer)
log52+log53=log5(2×3) ← 1.
=log56

(2)log745−log75

(Answer)
log745−log75=log7 ← 2.
=log79

(3)log450−log46+log415

(Answer)
log450−log46+log415=log4 ← 1., 2.
=log4125=log453 ← 3.
=3 log45

(4)log10 +log10

(Answer)

log10 +log10 =log10 ← 1.

=log1027=log1033 ← 3.
=3 log103

(5)log5

(Answer)

log5 =log5 4 ← (Root←→Exponent)

=log5 (22) ← Exponent Rule

=log5 2 ← 3.

=log52

(Root←→Exponent)=a
Exponent Rule(am)n=amn

(6)3 log10 +log10

(Answer)

3 log10 +log10 ← 3.

=log10 +log10 ← 1.

=log10 =log10

=log101−log102 ← I

=0−log102=−log102

Example 2Evaluate each of the following logarithms.

(1)log62+log63

(Answer)
log62+log63=log6(2×3) ← 1.
=log66=1 ← II

(2)log345−log35

(Answer)
log345−log35=log3 ← 2.
=log39
=log3(32) ← 3.
=2 log33 ← II
=2

(3)log450−log46+log415

(Answer)
log550−log56+log515=log5 ← 1., 2.
=log5125=log553 ← 3.
=3 log55 ← II
=3

(4)log2 −log2

(Answer)

log2 −log2 =log2 ← 2.

=log28=log223 ← 3.
=3 log22 ← II
=3

(5)log55−log56+log5

(Answer)

log55−log56+log5 ← 3.

=log55−log5+log5 ← 1.

=log5 =log5

=log51−log55 ← I, II

=0−1=−1

Question 1Simplify each of the following logarithms.
(Fill in the blanks.)

Question 2Evaluate each of the following logarithms.
(Fill in the blanks.)