Worksheet: Laws of Logarithms

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

Q1:

Calculate loglog1923.

Q2:

Find the value of logloglog1,280225 without using a calculator.

Q3:

Simplify loglog16×243.

Q4:

Determine the value of (3)93100loglogloglog without using a calculator.

  • A l o g 3
  • B 2 3 l o g
  • C3
  • D 1 3 l o g
  • E l o g 3 2

Q5:

What is the value of log(0.2)?

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

Q6:

Without using a calculator, find the approximate value of log5243 given that log51.465.

Q7:

Find the approximate value of log20, given that log51.161.

Q8:

Given that log30.5646, find, without using a calculator, the value of log147 correct to four decimal places.

Q9:

Determine the value of logloglog71005623+2523 without using a calculator.

Q10:

Given that 𝑥=5+26, find the value of log1𝑥+𝑥.

Q11:

Find log27.

Q12:

Find the value of logloglogloglog132+142+161264 without using a calculator.

Q13:

If 𝑎(1,1,000), what interval contains log𝑎?

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

Q14:

Given that log70.8451 and log20.301, round the value of log224 without using a calculator.

  • A2.3502
  • B1.1461
  • C 0 . 6 6 0 1
  • D4.5265
  • E0.6601

Q15:

Find the value of 5×89 correct to four decimal places.

Q16:

Find the value of logloglogloglog42+47+6629476 without using a calculator.

Q17:

Find the value of loglogloglog512×729×256×27, giving your answer in its simplest form.

Q18:

Given that log2=𝑥 and log9=𝑦, find log72 in terms of 𝑥 and 𝑦.

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

Q19:

Calculate loglogloglog0.08124316+1558112.

Q20:

Given that 𝜃0,𝜋2, evaluate logcoslogsec(𝜃)+(𝜃).

Q21:

Find the value of 225+12+18+7280loglogloglog in its simplest form.

Q22:

Simplify 𝑒ln completely.

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

Q23:

Evaluate logloglog𝑎+𝑏+𝑐.

  • A 𝑎 + 𝑏 + 𝑐
  • B 𝑎 𝑏 𝑐
  • C1
  • D l o g 𝑎 𝑏 𝑐

Q24:

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

  • A1
  • B 𝑥
  • C 𝑥
  • D 𝑥

Q25:

Is it true that logloglog(𝑥+𝑦)=𝑥+𝑦?

  • Ano
  • Byes

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