EVALUATING LOGARITHMIC EXPRESSIONS

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

Example 1 :

Evaluate :

log₃27 + log₃729

Solution :

=  log₃27 + log₃729

=  log₃3³ + log₃3⁶

=  3log₃3 + 6log₃3

=  3(1) + 6(1)

=  3 + 6

=  9

Example 2 :

Simplify :

log₁₀8 + log₁₀5 - log₁₀4

Solution :

=  log₁₀8 + log₁₀5 - log₁₀4

=  log₁₀ (8 ⋅ 5)/4]

=  log₁₀10

=  1

Example 3 :

Evaluate :

log₇98 - log₇14 + log₇343

Solution :

=  log₇98 - log₇14 + log₇343

=  log₇(98/14) + log₇343

=  log₇7 + log₇7³

=  1 + 3log₇7

=  1 + 3(1)

=  1 + 3

=  4

Example 4 :

Evaluate :

(1/2)log₉36 + 2log₉4 - 3log₉4

Solution :

=  (1/2)log₉36 + 2log₉4 - 3log₉4

=  log₉(36)^1/2 + log₉4² - log₉4³

=  log₉√36 + log₉16 - log₉64

=  log₉6 + log₉16 - log₉64

=  log₉(6 ⋅ 16) - log₉64

=  log₉96 - log₉64

=  log₉(96/64)

=  log₉(3/2)

Example 5 :

Simplify :

log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7

Solution :

In the given expression, logarithms have bases.

First group the logarithms with the same base and simplify.    

=  (log₃2 ⋅ log₄3) ⋅ (log₅4 ⋅ log₆5) ⋅  (log₇6 ⋅ log₈7)

=  log₄2 ⋅ log₆4 ⋅ log₈6

=  log₆2 ⋅ log₈6

=  log₈2

=  1/log₂8

=  1/log₂2³

=  1/3(log₂2)

=  1/3(1)

=  1/3

Example 6 :

Simplify :

log₇21 + log₇77 + log₇88 - log₇121 - log₇24

Solution :

=  log₇21 + log₇77 + log₇88 - log₇121 - log₇24

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

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

=  log₇ 142296 - log₇2904

=  log₇(142296 / 2904)

=  log₇49

=  log₇7²

=  2log₇7

=  2(1)

=  2

Example 7 :

Simplify :

log₈16 + log₈52 - 1/log₁₃8

Solution :

=  log₈16 + log₈52 - 1/log₁₃8

=  log₈16 + log₈52 - log₈13

=  log₈[(16 ⋅ 52)/13]

=  log₈(16 ⋅ 4)

=  log₈64

=  log₈8²

=  2log₈8

=  2(1)

=  2

Example 8 :

Simplify : 

5log₁₀2 + 2log₁₀3 - 6log₆₄4

Solution :

=  5log₁₀2 + 2log₁₀3 - (2⋅3) log₆₄4

=  log₁₀2⁵ + log₁₀3² - 2log₆₄4³

=  log₁₀32 + log₁₀9 - 2log₆₄64

=  log₁₀32 + log₁₀9 - 2(1)

=  log₁₀32 + log₁₀9 - 2log₁₀10

=  log₁₀(32 ⋅ 9) - log₁₀10²

=  log₁₀288 - log₁₀100

=  log₁₀(288/100)

=  log₁₀(72/25)

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