Worksheet: Logarithmic Equations with Like Bases

In this worksheet, we will practice solving logarithmic equations with like bases using the laws of exponents and logarithms.

Q1:

Determine the solution set of the equation logloglogīŠŽīŠŽīŠŽ(đ‘Ĩ−6)+(đ‘Ĩ+6)=64 in ℝ.

  • A { 1 0 }
  • B { 2 }
  • C { 8 }
  • D { 2 , − 2 }
  • E { 1 0 , − 1 0 }

Q2:

Find đ‘Ĩ such that log(4đ‘Ĩ−4)=2.

  • A26
  • B − 2 6
  • C24
  • D − 2 4

Q3:

Solve the equation logī€šđ‘Ĩ−đ‘Ĩ−2ī…=1īŠ¨, where đ‘Ĩ∈ℝ.

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

Q4:

Find the solution set of loglogīŠŦīŠŦ(−2đ‘Ĩ+48)=2đ‘Ĩ in ℝ.

  • A { 6 }
  • B { 1 4 }
  • C { 8 }
  • D { 5 }
  • E { 7 }

Q5:

Solve logīŠĒīŠ¨đ‘Ĩ−26đ‘Ĩ−7đ‘Ĩ+6=1, where đ‘Ĩ∈ℝ.

  • A { 1 , 4 }
  • B { − 6 , 4 }
  • C { 6 , − 4 }
  • D { − 6 }

Q6:

Find the solution set of ī€ēđ‘Ĩī†+đ‘Ĩ+1=ī€ē2ī†logloglogīŠ­īŠ¨īŠ­īŠ¨īŠ­īŠ¨ in ℝ.

  • A īŦ 2 7 , 7 ī¸
  • B īŦ 2 7 , 1 1 4 ī¸
  • C { 4 9 , 1 4 }
  • D { 1 4 , − 1 4 }
  • E īŦ 7 2 , 1 4 ī¸

Q7:

Determine the solution set of the equation loglogīŠ¨īŠ¨đ‘Ĩ=4−(đ‘Ĩ+6) in ℝ.

  • A { 1 0 }
  • B { 6 }
  • C { 2 }
  • D { − 1 }
  • E { 8 }

Q8:

What is the solution set of the equation logī—(9đ‘Ĩ−18)=2?

  • A ∅
  • B { 2 }
  • C { 3 , 6 }
  • D īŦ 2 0 9 ī¸

Q9:

Find the solution set of logloglogīŠŦīŠ¨īŠŦīŠŦī€š9đ‘Ĩ−27ī…=2đ‘Ĩ+13216 in ℝ.

  • A { 4 }
  • B { 9 }
  • C īŦ 3 3 8 ī¸
  • D { 6 }
  • E { 3 }

Q10:

Find the solution set of loglogloglogī—ī—ī—ī—5+40−24=2+8 in ℝ.

  • A īŦ 5 4 ī¸
  • B { 2 }
  • C īŦ 1 4 ī¸
  • D īŦ 4 5 ī¸

Q11:

Determine the solution set of the equation 2×5=64,000īŠŠī—ī—loglogīŽĸīŽĸ in ℝ.

  • A { 3 }
  • B { 8 }
  • C { 2 7 }
  • D { 4 0 }

Q12:

Find the solution set of loglogđ‘Ĩ2×2đ‘Ĩ=−4 in ℝ.

  • A { 2 0 , 1 0 }
  • B īŦ 1 0 0 , 1 1 0 0 ī¸
  • C īŦ 2 0 , 1 5 0 ī¸
  • D īŦ 2 0 0 , 1 5 0 ī¸
  • E { 2 0 0 , 1 0 0 }

Q13:

Find the solution set of loglogloglogīŠ¯īŠ¯īŠ¯īŠ¯īŠŦ216×81=36×đ‘Ĩ in ℝ.

  • A { 2 }
  • B { 9 }
  • C { 8 1 }
  • D { 2 7 }
  • E { 3 }

Q14:

Solve 5=15logīŽĨī—, where đ‘Ĩ∈ℝ.

  • A6
  • B1
  • C 1 6
  • D 6 5

Q15:

Solve logīŠ¨ī—(2+510)=10−đ‘Ĩ, where đ‘Ĩ∈ℝ.

Q16:

Determine the solution set of the equation loglogīŠŠīŠ¨īŠŠī€šđ‘Ĩ−5đ‘Ĩ+4ī…=4+(đ‘Ĩ−1) in ℝ.

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

Q17:

Solve the equation logī—īŠŠīŠĢđ‘Ĩ=5đ‘Ĩ, where đ‘Ĩ∈ℝ.

  • A30
  • B7
  • C10
  • D 1 7
  • E175

Q18:

If 8đ‘Ĩ+5đ‘Ļ−đ‘Ĩđ‘Ļ=3logloglogīŠŦīŠŦīŠŦīŠŦīŠŠ, what is the value of đ‘Ĩđ‘Ļ?

  • A 4 0 √ 6
  • B6
  • C 6 √ 6
  • D36

Q19:

Determine the solution set of ī„Ÿī€ž503+3īŠ=8loglogīŠŽī„īŠŽâˆšīŠŠī‘īŽĸ in ℝ.

  • A { 4 }
  • B { 6 }
  • C { 1 2 }
  • D { 9 }

Q20:

Find the value of đ‘Ĩ for which logī—īŠąīŠ§243√3=112.

Q21:

Find the solution set of loglogīŠŠī—īŠŠ17+đ‘Ĩ+5=đ‘Ĩī€ē1−21ī† in ℝ.

  • A { − 2 }
  • B { 2 7 }
  • C { − 5 }
  • D { 3 4 3 }
  • E { − 1 5 }

Q22:

Find the solution set of loglogīŠĒīŠĒī€ē64đ‘Ĩī†=4 in ℝ.

  • A { 7 }
  • B { 2 5 6 }
  • C { 4 }
  • D { 1 }

Q23:

Given that 10đ‘Ĩ+12đ‘Ļ−đ‘Ĩđ‘Ļ=7(7+10)logloglogloglogīŠŠīŠĢ, determine the value of đ‘Ĩ.

  • A 7 0 đ‘Ļ
  • B 8 đ‘Ļ
  • C 7 đ‘Ļ
  • D 4 9 đ‘Ļ
  • E 1 7 đ‘Ļ

Q24:

Find the solution set of īŽ§īŽ§ī„đ‘Ĩ=10log√ī— in ℝ.

  • A { 6 4 }
  • B { 8 , − 8 }
  • C īŠ 1 0 , 1 0 īĩ īŠŽ īŠą īŠŽ
  • D { 3 , − 3 }
  • E { 1 , 0 0 0 , 0 . 0 0 1 }

Q25:

Find the value of đ‘Ĩ for which logī—243=−5.

  • A − 1 5
  • B 1 3
  • C − 5 3
  • D8
  • E − 3 5

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