Worksheet: Laws of Logarithms

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In this worksheet, we will practice using the laws of logarithms to simplify logarithmic expressions.

Q1:

Calculate l o g l o g 1 9 2 − 3 .

Q2:

Find the value of l o g 1 , 2 8 0 − 2 l o g 2 − l o g 5 without using a calculator.

Q3:

Simplify l o g l o g 1 6 × 2 4 3 .

Q4:

Determine the value of ( 3 ) − 9 3 − 1 0 0 l o g l o g l o g l o g without using a calculator.

  • A 1 3 l o g
  • B 2 3 l o g
  • C l o g 3 2
  • D l o g 3
  • E3

Q5:

What is the value of l o g ( 0 . 2 ) ?

  • A − 2 3
  • B 3 2
  • C 2 3
  • D − 3 2
  • E − 5

Q6:

Without using a calculator, find the approximate value of l o g 5 2 4 3 given that l o g 5 ≈ 1 . 4 6 5 .

Q7:

Find the approximate value of l o g 2 0 , given that l o g 5 ≈ 1 . 1 6 1 .

Q8:

Given that l o g 3 ≈ 0 . 5 6 4 6 , find, without using a calculator, the value of l o g 1 4 7 correct to four decimal places.

Q9:

Determine the value of l o g l o g l o g 7 1 0 0 − 5 6 2 3 + 2 5 2 3 without using a calculator.

Q10:

Given that 𝑥 = 5 + 2 √ 6 , find the value of l o g 1 𝑥 + 𝑥 .

Q11:

Find l o g 3 √ 3 2 7 .

Q12:

Find the value of l o g l o g l o g l o g l o g 1 3 2 + 1 4 2 + 1 6 − 1 2 − 6 4 without using a calculator.

Q13:

If 𝑎 ∈ ( 1 , 1 , 0 0 0 ) , what interval contains l o g 𝑎 ?

  • A ( 3 , ∞ )
  • B ( − ∞ , 0 )
  • C ( 0 , 3 ]
  • D ( 0 , 3 )
  • E [ 3 , ∞ )

Q14:

Given that l o g 7 ≈ 0 . 8 4 5 1 and l o g 2 ≈ 0 . 3 0 1 , round the value of l o g 2 2 4 without using a calculator.

  • A0.6601
  • B − 0 . 6 6 0 1
  • C1.1461
  • D2.3502
  • E4.5265

Q15:

Find the value of 5 × 8 9 correct to four decimal places.

Q16:

Find the value of l o g l o g l o g l o g l o g 4 2 + 4 7 + 6 √ 6 2 9 4 − 7 √ 6 without using a calculator.

Q17:

Find the value of l o g l o g l o g l o g 5 1 2 × 7 2 9 × 2 5 6 × 2 7 , giving your answer in its simplest form.

Q18:

Given that l o g 2 = 𝑥 and l o g 9 = 𝑦 , find l o g 7 2 in terms of 𝑥 and 𝑦 .

  • A 8 𝑦
  • B 𝑥 + 3 𝑦
  • C 2 7 𝑥
  • D 3 𝑥 + 𝑦
  • E 4 𝑥 + 𝑦

Q19:

Calculate l o g l o g l o g l o g 0 . 0 8 1 − 2 4 3 1 6 + 1 5 5 8 − 1 1 2 .

Q20:

Given that 𝜃 ∈ 0 , 𝜋 2 , evaluate l o g c o s l o g s e c ( 𝜃 ) + ( 𝜃 ) .

Q21:

Find the value of 2 2 5 + 1 2 + 1 8 + 7 2 − 8 0 l o g l o g l o g l o g in its simplest form.

Q22:

Simplify 𝑒 l n ( ) 1 2 completely.

  • A l n 1 2
  • B 1 4
  • C 𝑒 ( ) 1 2
  • D 1 2
  • E 𝑒 2

Q23:

Evaluate l o g l o g l o g 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 𝑏 𝑐 𝑎 + 𝑏 + 𝑐 .

  • A l o g 𝑎 𝑏 𝑐
  • B 𝑎 𝑏 𝑐
  • C 𝑎 + 𝑏 + 𝑐
  • D1

Q24:

If 𝑥 is any real, positive number except 1, which of the following is equal to l o g 𝑥 𝑥 𝑥 ?

  • A 𝑥 3
  • B 𝑥 2
  • C1
  • D 𝑥

Q25:

Is it true that l o g l o g l o g 𝑎 𝑎 𝑎 ( 𝑥 + 𝑦 ) = 𝑥 + 𝑦 ?

  • Ano
  • Byes

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