# EVALUATE BASIC LOG FUNCTIONS WITHOUT CALCULATOR

Some of the rules used in logarithms,

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

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

log mn = n log m

loga a = 1

Without using a calculator, simplify.

Problem 1 :

log 8 / log 2

Solution :

Simplifying the numerator and denominators separately.

log 8 = log23

= 3log2

3log2/log2

= 3(1)

= 3

Problem 2 :

log 9 / log 3

Solution :

Simplifying the numerator, we get

= log 9 / log 3

= log 32

= 2 log 3

Applying it in the given question, we get

= 2 log 3/log3

= 2(1)

= 2

Problem 3 :

log 4 / log 8

Solution :

= log 4 / log 8

= log 22/log 23

= 2/3

Problem 4 :

log 5 / log (1/5)

Solution :

= log 5 / log (1/5)

= log 5 / log 5-1

= log 5 / -1log 5

= -(log 5/log 5)

= -1

Problem 5 :

log (0.5) / log 2

Solution :

= log (0.5) / log 2

= log (5/10)/ log 2

log (5/10) x 2

= log5/5

= 1

Problem 6 :

log 8 / log (0.25)

Solution :

= log 8 / log (0.25)

log 8 = log 23

= 3 log 2 ----(1)

log (0.25) = log (25/100)

= log 1/4

= log 1/22

= log 2-2

= -2 log 2 ----(2)

(1) / (2)

= 3 log 2/(-2 log 2)

= -3/2

Problem 7 :

log 2b / log 8

Solution :

logb a = log a / log b

= log 2b / log 8

= log8 2b

= b × log8 2

= b × log23 2

= b × 1/3 × log22

= b/3

Problem 8 :

log 4 / log 2a

Solution :

= log 4 / log 2a

log 4 = log 22

= 2 log 2 ---(1)

log 2a = a log 2 ---(2)

(1) / (2)

= 2 log 2/a log 2

= 2/a

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