EVALUATE LOGARITHMS WITHOUT USING CALCULATOR

To solve a logarithmic equation, first combine the logarithmic terms using rules of logarithms.

Some of the rules in logarithm :

log m + log n = log (m x n)

log m - log n = log (m / n)

log mn = n log m

log a a = 1

log a b = 1/log b a

Find :

Problem 1 :

log9

Solution :

= log9

= log3 32

= 2 log3 3

= 2 (1)

= 2

Problem 2 :

log32

Solution :

= log32

= log2 25

= 5 log2 2

= 5 (1)

= 5

Problem 3 :

log√2

Solution :

= log√2

= log2 21/2

= 1/2 log2 2

= 1/2 (1)

= 1/2

Problem 4 :

log√2

Solution :

= log√2

= log4 √2= log421/2= 12log42= 12 log24 = 12 log222= 12(2) log22= 14(1)= 14

Problem 5 :

log3√3

Solution :

= log3√3

= log3 3√3= log331+1/2= log333/2= 32log33= 32(1)= 32

Problem 6 :

log6 1

Solution :

= log6 1

= 0

Problem 7 :

log8 8

Solution :

= log8 8

= 1

Problem 8 :

log8 (1/8)

Solution :

= log8 (1/8)

= log8 8-1

= (-1) log8 8

= -1 × 1

= -1

Problem 9 :

log1/8 (1/8)

Solution :

= log1/8 (1/8)

= 1

Problem 10 :

log√2 (1/√2)

Solution :

= log√2 (1/√2)

= log√2 √2-1

= (-1) log√2 √2

= -1 × 1

= -1

Problem 11 :

log2 (1/√2)

Solution :

= log2 (1/√2)

= log2 2-1/2

= (-1/2) log2 2

= -1/2 × 1

= -1/2

Problem 12 :

log8 (1/√2)

Solution :

= log8 (1/√2)

= log8 (1/2)1/2

= log(2)-1/2

= -12log82= -12log28= -12log223= -12(3log22)= -16(1)= -16

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