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

Video Lessons

Introduction to Logarithms

Fundamental Laws of Logarithms

Change of Base

Using Log Table

Using Antilog Table

Important Stuff

Difference between Bar Value and Negative Value

Multiplication of Two Logarithms

Relationship between Exponents and Logarithms

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