Worksheet: Properties of Logarithms

In this worksheet, we will practice using the product, quotient, and power rules of logarithms to evaluate expressions and solve equations.

Q1:

If ๐‘ฅ is any positive, real number, which of the following is equal to l o g ๐‘ฅ ๐‘› ?

  • A l o g ๐‘ฅ
  • B ๐‘›
  • C l o g ๐‘ฅ ๐‘›
  • D ๐‘› ๐‘ฅ l o g

Q2:

Which of the following is equal to l o g ๐‘Ž ๏€ฝ ๐‘ฅ ๐‘ฆ ๏‰ ?

  • A l o g l o g ๐‘Ž ๐‘Ž ๐‘ฅ รท ๐‘ฆ
  • B l o g l o g ๐‘Ž ๐‘Ž ๐‘ฅ ร— ๐‘ฆ
  • C l o g l o g ๐‘Ž ๐‘Ž ๐‘ฅ + ๐‘ฆ
  • D l o g l o g ๐‘Ž ๐‘Ž โˆ’ 1 ๐‘ฅ + ๐‘ฆ

Q3:

If l o g ๏Šฏ ๐‘ฅ + l o g ๏Šฏ 9 = 2 , what is the value of ๐‘ฅ ?

Q4:

Find the value of l o g l o g l o g l o g ๏Šญ ๏Šญ ๏Šญ ๏Šญ 3 2 + 8 1 0 โˆ’ 5 without using a calculator.

Q5:

Find the value of l o g l o g l o g l o g ๏Šฌ ๏Šฌ ๏Šฌ ๏Šฌ 4 + 2 7 ร— 8 1 7 2 9 without using a calculator.

Q6:

Is it true that l o g l o g l o g ๐‘Ž ๐‘Ž ๐‘Ž ( ๐‘ฅ + ๐‘ฆ ) = ๐‘ฅ ร— ๐‘ฆ ?

  • Ano
  • Byes

Q7:

Which of the following is equal to l o g ๐‘Ž ( ๐‘ฅ ๐‘ฆ ) ?

  • A l o g l o g ๐‘Ž ๐‘Ž ๐‘ฅ โˆ’ ๐‘ฆ
  • B l o g l o g ๐‘Ž ๐‘Ž ๐‘ฅ ร— ๐‘ฆ
  • C l o g l o g ๐‘Ž โˆ’ 1 ๐‘Ž โˆ’ 1 ๐‘ฅ + ๐‘ฆ
  • D l o g l o g ๐‘Ž ๐‘Ž ๐‘ฅ + ๐‘ฆ

Q8:

If l o g l o g l o g 6 6 6 ๐‘ฅ โˆ’ 7 = 1 0 , what is the value of ๐‘ฅ ?

Q9:

Find the solution set of l o g l o g l o g 9 2 9 9 ๐‘ฅ + 2 = 3 2 in โ„ .

  • A { 9 , โˆ’ 9 }
  • B { 8 , โˆ’ 8 }
  • C { 1 6 , โˆ’ 1 6 }
  • D { 4 , โˆ’ 4 }

Q10:

Find the solution set of l o g l o g l o g ( ๐‘ฅ + 8 ) + ( ๐‘ฅ โˆ’ 8 ) = ( 1 9 ๐‘ฅ + 2 ) in โ„ .

  • A { 8 }
  • B { โˆ’ 3 }
  • C { 1 0 }
  • D { 2 2 }
  • E { 1 9 }

Q11:

Calculate l o g l o g l o g l o g 1 6 โˆ’ 2 5 6 4 โˆ’ 1 2 5 .

  • A9
  • B2
  • C 3 2
  • D 2 3
  • E5

Q12:

Find the perimeter of the figure below.

Q13:

Determine the perimeter of the figure below.

Q14:

Simplify l o g l o g 7 7 1 6 2 5 6 .

  • A12
  • B4
  • C2
  • D 1 2
  • E5

Q15:

Find the value of 5 9 โˆš 3 l o g ๏Šฉ ๏Žค without using a calculator.

Q16:

Find the value of 1 8 ๐‘Ž + 1 8 ๐‘ + 5 5 ๐‘ โˆ’ โˆš ๐‘Ž ๐‘ โˆ’ 5 ๐‘ l o g l o g l o g l o g l o g ๏Šซ ๏Šซ ๏Šซ ๏Šซ ๏Šซ ๏Šซ ๏Žง .

Q17:

Find the solution set of l o g l o g l o g ( ๐‘ฅ + 5 ) + ( ๐‘ฅ โˆ’ 5 ) = 3 โˆ’ 5 in โ„ .

  • A { 6 }
  • B { 2 3 }
  • C { 2 0 0 }
  • D { 1 5 }
  • E { 5 }

Q18:

Find the solution set of l o g l o g l o g 8 2 8 6 ๏€น ๐‘ฅ + 5 0 ๐‘ฅ โˆ’ 7 9 ๏… โˆ’ ( ๐‘ฅ โˆ’ 2 ) = 3 6 in โ„ .

  • A { 6 }
  • B { 2 }
  • C { 1 0 }
  • D { 7 }

Q19:

Find the solution set of 4 ( ๐‘ฅ + 5 ) โˆ’ ( ๐‘ฅ โˆ’ 3 ) = 6 2 5 l o g l o g l o g ๏Šจ ๏Šจ ๏Šช ๏Šจ in โ„ .

  • A { 1 1 }
  • B { 2 }
  • C { 7 }
  • D { 5 }

Q20:

Find the solution set of ๏€บ ๐‘ฅ ๏† = ๐‘ฅ l o g l o g 4 3 4 4 in โ„ .

  • A ๏ฌ 1 6 , 1 4 ๏ธ
  • B ๏ฌ 6 4 , 1 6 4 ๏ธ
  • C { 1 6 }
  • D ๏ฌ 1 6 , 1 1 6 ๏ธ
  • E ๏ฌ 4 , 1 1 6 ๏ธ

Q21:

Find the solution set of ๏€บ ๐‘ฅ ๏† โˆ’ ๐‘ฅ = 8 l o g l o g 2 2 2 โˆ’ 7 in โ„ .

  • A ๏ฌ 2 5 6 , 1 1 2 8 ๏ธ
  • B ๏ฌ 2 5 6 , 1 2 ๏ธ
  • C { 2 }
  • D ๏ฌ 2 , 1 2 5 6 ๏ธ
  • E ๏ฌ 1 2 8 , 1 2 5 6 ๏ธ

Q22:

Find the solution set of l o g l o g l o g l o g 9 9 2 9 9 2 7 2 9 ๐‘ฅ = ๏€บ 2 ๏† โˆ’ 8 ๏€ป ๏‡ in โ„ .

  • A { 1 }
  • B { 3 }
  • C { 4 }
  • D { 2 }
  • E { 8 1 }

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