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 = logxa ⋅ logbx
logba = logxa / logxb
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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