Worksheet: Logarithmic Equations with Different Bases

In this worksheet, we will practice solving logarithmic equations involving logarithms with different bases.

Q1:

Find the solution set of l o g l o g   π‘₯ = 4 in ℝ .

  • A { 5 }
  • B { 2 }
  • C { 3 }
  • D { 4 }

Q2:

Select the expression equal to l o g l o g   π‘₯ 𝑦 .

  • A l o g l o g   π‘Ž π‘Ž
  • B l o g l o g   π‘₯ 𝑦
  • C l o g l o g   𝑦 π‘₯
  • D l o g l o g   𝑏 𝑏

Q3:

Find the solution set of l o g l o g   π‘₯ + 9 2 = 6 in ℝ .

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

Q4:

Determine the solution set of the equation l o g l o g οŠͺ  π‘₯ + 2 5 4 = 1 0 in ℝ .

  • A { 2 0 }
  • B { 1 0 }
  • C { 1 , 0 2 4 }
  • D { 5 }
  • E { 9 }

Q5:

Solve l o g l o g   π‘₯ = 1 4 9 , where π‘₯ ∈ ℝ .

  • A βˆ’ 1 4
  • B2
  • C4
  • D 1 4
  • E βˆ’ 4

Q6:

Find the solution set of l o g ( 1 0 βˆ’ 9 0 ) + π‘₯ βˆ’ 3 = 0  in ℝ .

  • A { 2 , 1 }
  • B { 2 }
  • C { 3 , 1 }
  • D { 1 }
  • E { 3 }

Q7:

Determine the solution set of the equation 3 6 βˆ’ 4 0 Γ— 6 + 1 4 4 = 0   . Give the values correct to two decimal places.

  • A { 4 . 0 0 , 0 . 3 9 }
  • B { 2 . 0 0 , 0 . 7 7 }
  • C { 4 . 0 0 , 1 . 2 9 }
  • D { 6 . 0 0 , 0 . 7 7 }
  • E { 2 . 0 0 , 1 . 2 9 }

Q8:

Find the solution set of 2 + 2 = 7 2     in ℝ .

  • A { βˆ’ 6 , βˆ’ 3 }
  • B { 6 , βˆ’ 3 }
  • C { βˆ’ 6 , 3 }
  • D { 6 , 3 }

Q9:

Solve ο€Ί π‘₯  βˆ’ 6 π‘₯ + 8 = 0 l o g l o g    , where π‘₯ ∈ ℝ .

  • A { 2 5 , 6 2 5 }
  • B  1 1 0 , 1 2 0 
  • C { 5 , 1 2 5 }
  • D { 1 0 , 2 0 }
  • E  1 2 5 , 1 6 2 5 

Q10:

Determine the solution set of the equation l o g l o g   οŠͺ   π‘₯ + π‘₯ + 3 = 0 in ℝ .

  • A  1 2 7 
  • B  1 3 √ 3 
  • C { 2 7 }
  • D  3 √ 3 

Q11:

Find the solution set of l o g l o g  οŠͺ π‘₯ = ( 3 π‘₯ + 2 8 ) in ℝ .

  • A { 6 4 }
  • B { 7 }
  • C { 4 }
  • D { 1 2 8 }
  • E { 1 1 }

Q12:

Find the solution set of ο€Ί π‘₯ + 2  ο€» π‘₯ 1 6  = 5 l o g l o g οŠͺ οŠͺ in ℝ .

  • A  1 6 , 1 1 6 
  • B  4 , 1 4 
  • C { 6 4 , 1 6 }
  • D  6 4 , 1 6 4 

Q13:

Determine the solution set of the equation l o g l o g οŠͺ οŠͺ π‘₯ βˆ’ 1 π‘₯ = 8 3 in ℝ .

  • A { 7 }
  • B { 4 , 3 }
  • C  6 4 , 1 √ 4  
  • D { 6 4 , 1 6 }
  • E  1 6 , 1 √ 2 

Q14:

Find all possible values of π‘₯ for which π‘₯ = 6 4 π‘₯ l o g     .

  • A4 or 1 8
  • B8
  • C12 or 1 1 8
  • D32
  • E8 or 1 4

Q15:

Find the solution set of 2 Γ— 8 = 2 (  )  l o g l o g     in ℝ .

  • A { 2 , 8 }
  • B { 5 1 2 , 8 }
  • C { 6 4 , 1 }
  • D { 3 , 1 }
  • E  4 3 

Q16:

Find the solution set of l o g l o g   π‘₯ βˆ’ 4 9 = βˆ’ 1 in ℝ .

  • A { 1 , βˆ’ 2 }
  • B { 1 }
  • C { 1 4 }
  • D  7 , 1 4 9 

Q17:

Find the solution set of l o g l o g     π‘₯ = 7 2 9 in ℝ .

  • A { 9 }
  • B { 8 1 }
  • C { 1 4 }
  • D { 5 }
  • E { 7 2 9 }

Q18:

Find the solution set of l o g l o g οŠͺ  π‘₯ + 2 5 4 = 1 0 in ℝ .

  • A { 2 5 6 }
  • B { 4 }
  • C { 1 , 0 2 4 }
  • D { 5 }

Q19:

Find the solution set of l o g l o g   π‘₯ + 2 5 2 = 1 0 in ℝ .

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

Q20:

Solve l o g l o g  οŠͺ π‘₯ = 6 4 , where π‘₯ ∈ ℝ .

  • A 1 2 7
  • B27
  • C 1 7 2 9
  • D729
  • E81

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