SIMPLIFYING LOGARITHMIC EXPRESSIONS

To simplify logarithmic expressions, you must be aware of the following three fundamentals laws of logarithms. 

Law 1 : 

Logarithm of product of two numbers is equal to the sum of the logarithms of the numbers to the same base. 

That is, 

log_amn  =  log_am + log_an

Law 2 : 

Logarithm of the quotient of two numbers is equal to the difference of their logarithms to the same base. 

That is, 

log_a(m/n)  =  log_am - log_an

Law 3 : 

Logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number to the same base.  

That is, 

log_amⁿ  =  nlog_am

Change of Base : 

log_ba  =  log_xa ⋅ log_bx

log_ba  =  log_xa / log_xb

Solved Examples

Example 1 :

Simplify : 

log₅ 25 +  log₅ 625

Solution :

=  log₅ 25 +  log₅ 625 

=  log₅ (25 ⋅ 625)

=  log₅ (5² ⋅ 5⁴)

=  log₅ 5^{(2 + 4)}

=  log₅ 5⁶

=  6log₅ 5

=  6 (1)

=  6

Example 2 :

Simplify : 

log₅ 4 +  log₅ (1/100)

Solution :

=  log₅ 4 +  log₅ (1/100) 

=  log₅ (4 ⋅ (1/100))

=  log₅ (1/25)

=  log₅ 5^-2

=  -2log₅ 5

=  -2(1)

=  -2

Example 3 :

Simplify :

log₈ 128 -  log₈ 16

Solution :

=  log₈ 128 -  log₈ 16

=  log₈ (128/16)

=  log₈ 8

=  1

Example 4 :

Simplify :

log₃ 2 ⋅ log₄ 3 ⋅ log₅ 4 ⋅ log₆ 5 ⋅ log₇ 6 ⋅ log₈ 7

Solution :

In the given expression, logarithms have bases.

First group the logarithms with the same base and simplify.    

=  (log₃ 2 ⋅ log₄ 3) ⋅ (log₅ 4 ⋅ log₆ 5) ⋅  (log₇ 6 ⋅ log₈ 7)

=  log₄ 2 ⋅ log₆ 4 ⋅ log₈ 6

=  log₆ 2 ⋅ log₈ 6

=  log₈ 2

=  1/log₂ 8

=  1/log₂ 2³

=  1/3(log₂ 2)

=  1/3(1)

=  1/3

Example 5 :

Simplify :

log₇ 21 + log₇ 77 + log₇ 88 - log₇ 121 - log₇ 24

Solution :

=  log₇21 + log₇77 + log₇88 - log₇121 - log₇24

=  log₇(21 ⋅ 77 ⋅ 88) - (log₇121 + log₇24)

=  log₇(21 ⋅ 77 ⋅ 88) - log₇(121 ⋅ 24)

=  log₇ 142296 - log₇ 2904

=  log₇ (142296 / 2904)

=  log₇49

=  log₇7²

=  2log₇7

=  2(1)

=  2

Example 6 :

Simplify :

log₈ 16 + log₈ 52 - 1/log₁₃ 8

Solution :

=  log₈ 16 + log₈ 52 - 1/log₁₃ 8

=  log₈ 16 + log₈ 52 - log₈ 13

=  log₈ [(16 ⋅ 52)/13]

=  log₈ (16 ⋅ 4)

=  log₈ 64

=  log₈ 8²

=  2log₈ 8

=  2(1)

=  2

Example 7 :

Simplify : 

5log₁₀ 2 + 2log₁₀ 3 - 6log₆₄ 4

Solution :

=  5log₁₀ 2 + 2log₁₀ 3 - (2⋅3)log₆₄ 4

=  log₁₀ 2⁵ + log₁₀ 3² - 2log₆₄ 4³

=  log₁₀ 32 + log₁₀ 9 - 2log₆₄ 64

=   log₁₀ 32 + log₁₀ 9 - 2(1)

=  log₁₀ 32 + log₁₀ 9 - 2log₁₀10

=  log₁₀ (32 ⋅ 9) - log₁₀10²

=  log₁₀ (288/100)

=  log₁₀ (72/25)

Example 8 :

Simplify :

log₁₀ 8 + log₁₀ 5 - log₁₀ 4

Solution :

=  log₁₀ 8 + log₁₀ 5 - log₁₀ 4

=  log₁₀ [(8 ⋅ 5)/4]

=  log₁₀ 10

=  1

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