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Worksheet: Laws of Logarithms

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

Q1:

Calculate l o g l o g 2 2 1 9 2 − 3 .

Q2:

Calculate l o g l o g 4 4 5 1 2 − 2 .

Q3:

Calculate l o g l o g 3 3 2 7 − 3 .

Q4:

Calculate l o g l o g 2 2 7 6 8 − 3 .

Q5:

Find the value of l o g l o g l o g 4 4 4 1 2 8 0 − 2 2 − 5 without using a calculator.

Q6:

Find the value of l o g l o g l o g 2 2 2 6 4 0 − 2 4 − 5 without using a calculator.

Q7:

Determine the value of ( 3 ) − 9 3 − 1 0 0 l o g l o g l o g l o g 2 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

Q8:

What is the value of l o g 0 . 0 4 − 3 ( 0 . 2 ) ?

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

Q9:

Simplify l o g l o g 3 2 1 6 × 2 4 3 .

Q10:

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

Q11:

Find the approximate value of l o g 4 2 0 , given that l o g 4 5 ≈ 1 . 1 6 1 .

Q12:

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

Q13:

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

Q14:

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

Q15:

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

Q16:

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

Q17:

Find l o g 3 √ 3 2 7 .

Q18:

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

Q19:

If , what interval contains ?

  • A
  • B
  • C
  • D
  • E

Q20:

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

Q21:

Find the value of 5 × 8 9 1 5 0 2 0 0 3 0 0 correct to four decimal places.

Q22:

Given that , evaluate .

Q23:

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 2 5 2 5 2 5 2 5 in its simplest form.

Q24:

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

Q25:

Find the value of l o g l o g l o g l o g 4 2 3 9 5 1 2 × 7 2 9 × 2 5 6 × 2 7 , giving your answer in its simplest form.

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