SIMPLIFYING LOGARITHMIC EXPRESSIONS

Simplify each of the following expressions :

Example 1 :

log₅ 25 +  log₅ 625

Solution :

=  log₅25 + log₅625 

=  log₅(25 ⋅ 625)

=  log₅(5² ⋅ 5⁴)

=  log₅5^{(2 + 4)}

=  log₅5⁶

 =  6log₅5

=  6(1)

=  6

Example 2 :

log₅ 4 +  log₅ (1/100)

Solution :

=  log₅4 + log₅(1/100) 

=  log₅(4 ⋅ 1/100)

=  log₅(1/25)

=  log₅(1/5²)

=  log₅5^-2

=  -2log₅5

=  -2(1)

=  -2

Example 3 :

log₈ 128 -  log₈ 16

Solution :

=  log₈128 -  log₈16

=  log₈(128/16)

=  log₈8

=  1

Example 4 :

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.

=  (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 5 :

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)

=  log₇142296 - log₇2904

=  log₇(142296/2904)

=  log₇49

=  log₇7²

=  2log₇7

=  2(1)

=  2

Example 6 :

log₈ 16 + log₈ 52 - 1/log₁₃ 8

Solution :

=  log₈16 + log₈52 - 1/log₁₃8

=  log₈16 + log₈52 - log₈13

=  log₈[(16 ⋅ 52)/13]

=  log₈(16 ⋅ 4)

=  log₈64

=  log₈8²

=  2log₈8

=  2(1)

=  2

Example 7 :

5log₁₀ 2 + 2log₁₀ 3 - 6log₆₄ 4

Solution :

=  5log₁₀2 + 2log₁₀3 - 6log₆₄4

=  5log₁₀2 + 2log₁₀3 - (2 ⋅ 3) log₆₄4

=  log₁₀ 2⁵ + log₁₀ 3² - 2log₆₄ 4³

=  log₁₀32 + log₁₀9 - 2log₆₄64

=  log₁₀32 + log₁₀9 - 2 (1)

=  log₁₀32 + log₁₀9 - 2log₁₀10

=  log₁₀(32 ⋅ 9) - log₁₀10²

=  log₁₀(288/100)

=  log₁₀(72/25)

Example 8 :

log₁₀ 8 + log₁₀ 5 - log₁₀ 4

Solution :

=  log₁₀8 + log₁₀5 - log₁₀4

=  log₁₀[(8 ⋅ 5)/4]

=  log₁₀(40/4)

=  log₁₀10

=  1

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