# 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,

logamn  =  logam + logan

Law 2 :

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

That is,

loga(m/n)  =  logam - logan

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,

logamn  =  nlogam

Change of Base :

logba  =  logx⋅ logbx

logba  =  logxa / logxb

## 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 -  log16

Solution :

=  log8 128 -  log16

=  log8 (128/16)

=  log8 8

=  1

Example 4 :

Simplify :

log3 2  log4⋅ log5⋅ log6⋅ log7⋅ 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)

=  log142296 - 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 25log10 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

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