LOGARITHM WORKSHEET

1. Find the logarithm of 32 to the base 2.

2. Find the value of log_2√31728.

3. Find the value of log(0.0001) to the base 0.1.

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

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

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

7. If log_a(√2) = 1/6, find the value of a.

8. Simplify : (1/2)log₁₀25 - 2log₁₀3 + log₁₀18.

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

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

11. 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)

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

13. If logx + logy = log(x + y), solve for y in terms of x.

Answers

1. Answer :

log₂32 = log₂(2)⁵

= 5log₂(2)

= 5(1)

= 5

2. Answer :

Write 1728 as a power of 2√3.

1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3

1728 = 2⁶ x 3³

1728 = 2⁶ x [(√3)²]³

1728 = 2⁶ x (√3)⁶

1728 = (2√3)⁶

log_2√3(1728) = log_2√3(2√3)⁶

Using the power rule of logarithms,

= 6log_2√3(2√3)

= 6(1)

= 6

3. Answer :

log_0.10.0001 = log_0.1(0.1)⁴

= 4log_0.10.1

= 4(1)

= 4

4. Answer :

log₉(1/81) = log₉1 - log₉81

= 0 - log₉(9)²

= -2log₉9

= -2(1)

= -2

5. Answer :

log₂(0.0625) = log₂(0.5)⁴

= 4log₂(0.5)

= 4log₂(1/2)

= 4(log₂1 - log₂2)

= 4(0 - 1)

= 4(-1)

= -4

6. Answer :

log₉(0.3) = log₉(1/3)

= log₉1 - log₉3

= 0 - log₉3

= -log₉3

= -1 / log₃9

= -1 / log₃3²

= -1 / 2log₃3

= -1 / 2(1)

= -1/2

7. Answer :

log_a(√2) = 1/6

Write the equation in exponential form.

√2 = a^1/6

Raise to the power 6 on both sides.

(√2)⁶ = (a^1/6)⁶

(2^1/2)⁶ = a

2³ = a

8 = a

8. Answer :

= (1/2)log₁₀25 - 2log₁₀3 + log₁₀18

Using power rule of logarithms,

= log₁₀25^1/2 - log₁₀3² + log₁₀18

= log₁₀(5²)^1/2 - log₁₀3² + log₁₀18

= log₁₀5 - log₁₀9 + log₁₀18

= log₁₀5 + log₁₀18 - log₁₀9

Using the product rule of logarithms,

= log₁₀(5 x 18) - log₁₀9

= log₁₀90 - log₁₀9

Using the quotient rule of logarithms,

= log₁₀(90/9)

= log₁₀10

= 1

9. Answer :

log6 = log(2 ⋅ 3)

= log2 + log3

Substitute the values of log2 and log3.

= 0.3010 + 0.4771

= 0.7781

10. Answer :

2logx = 4log3

Divide each side by 2.

logx = 2log3

logx = log3²

logx = log9

x = 9

11. Answer :

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

12. Answer :

1 + abc = 1 + log₂₄12 ⋅ log₃₆24 ⋅ log₄₈36

= 1 + log₃₆12 ⋅ log₄₈36

= 1 + log₄₈12

= log₄₈48 + log₄₈12

= log₄₈(48 ⋅ 12)

= log₄₈(2 ⋅ 12)²

= 2log₄₈24

= 2log₃₆24 ⋅ log₄₈36

= 2bc

13. Answer :

logx + logy = log(x + y)

Using the product rule of logarithms on the left side of the equation,

log(xy) = log(x + y)

xy = x + y

Subtract y from both sides.

xy - y = x

Factor.

y(x - 1) = x

Divide both sides by (x - 1).

y = x/(x - 1)

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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Problem - 22

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