LOGARITHM PROBLEMS

Problem 1 :

Find the logarithm of 64 to the base 2√2.

Solution :

Write 64 as in terms of 2√2.

64 = 2⁶

= 2⁴⁺²

= 2⁴ ⋅ 2²

= 2⁴ ⋅ [(√2)²]²

= 2⁴ ⋅ (√2)⁴

= (2√2)⁴

log_2√264 = log_2√2(2√2)⁴

= 4log_2√2(2√2)

= 4(1)

= 4

Problem 2 :

If log_abc = x, log_bca = y and log_cab = z, then find the value of

1/(x + 1) + 1/(y + 1) + 1/(z + 1)

Solution :

x + 1 = log_abc + log_aa = log_aabc

y + 1 = log_bca + log_bc = log_babc

z + 1 = log_cab + log_cc = log_cabc

1/(x + 1) = 1/log_aabc = log_abca

1/(y + 1) = 1/log_babc = log_abcb

1/(z + 1) = 1/log_cabc = log_abcc

1/(x + 1) + 1/(y + 1) + 1/(z + 1) = log_abca + log_abcb + log_abcc

= log_abcabc

= 1

Problem 3 :

If a = log₂₄12, b = log₃₆24 and c = log₄₈36, then find the value of (1 + abc) in terms of b and c.

Solution :

1 + abc = 1 + log₂₄12 ⋅ log₃₆24 ⋅ log₄₈36

= 1 + log₃₆12 ⋅ log₄₈36

= 1 + log₄₈12

= log₄₈48 + log₄₈12

= log₄₈(48 ⋅ 12)

= log₄₈(2 ⋅ 12)²

= 2log₄₈24

= 2log₃₆24 ⋅ log₄₈36

= 2bc

Problem 4 :

Find the value of log 0.0001 to the base 0.1.

Solution :

log_0.10.0001 = log_0.1(0.1)⁴

= 4log_0.10.1

= 4(1)

= 4

Problem 5 :

If 2logx = 4log3, then find the value of x.

Solution :

2logx = 4log3

Divide each side by 2.

logx = 2log3

logx = log3²

logx = log9

x = 9

Problem 6 :

Find the value of log_√264.

Solution :

log_√264 = log_√2(2)⁶

= 6log_√2(2)

= 6log_√2(√2)²

= 6 ⋅ 2log_√2(√2)

= 12 ⋅ 2(1)

= 12

Problem 7 :

Find the value of log (1/81) to the base 9.

Solution :

log₉(1/81) = log₉1 - log₉81

= 0 - log₉(9)²

= -2log₉9

= -2(1)

= -2

Problem 8 :

Find the value of log(0.0625) to the base 2.

Solution :

log₂(0.0625) = log₂(0.5)⁴

= 4log₂(0.5)

= 4log₂(1/2)

= 4(log₂1 - log₂2)

= 4(0 - 1)

= 4(-1)

= -4

Problem 9 :

Given log2 = 0.3010 and log3 = 0.4771, find the value of log6.

Solution :

log6 = log(2 ⋅ 3)

= log2 + log3

Substitute the values of log2 and log3.

= 0.3010 + 0.4771

= 0.7781

Problem 10 :

Find the value of log(0.3) to the base 9.

Solution :

log₉(0.3) = log₉(1/3)

= log₉1 - log₉3

= 0 - log₉3

= -log₉3

= -1 / log₃9

= -1 / log₃3²

= -1 / 2log₃3

= -1 / 2(1)

= -1/2

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