EVALUATE BASIC LOG FUNCTIONS WITHOUT CALCULATOR

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 2^b / log 8

Solution :

log_b a = log a / log b

= log 2^b / log 8

= log₈ 2^b

= b × log₈ 2

= b × log₂³ 2

= b × 1/3 × log₂2

= b/3

Problem 8 :

log 4 / log 2^a

Solution :

= log 4 / log 2^a

 log 4 = log 2²

= 2 log 2 ---(1)

log 2^a = a log 2 ---(2)

(1) / (2)

= 2 log 2/a log 2

= 2/a