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 loglog𝑥=4 in .

  • A{3}
  • B{2}
  • C{4}
  • D{5}

Q2:

Find the solution set of loglog𝑥+92=6 in .

  • A{3}
  • B{8}
  • C{2}
  • D{4}

Q3:

Determine the solution set of the equation loglog𝑥+254=10 in .

  • A{20}
  • B{10}
  • C{1,024}
  • D{9}
  • E{5}

Q4:

Solve loglog𝑥=149, where 𝑥.

  • A2
  • B4
  • C14
  • D14
  • E4

Q5:

Find the solution set of log(1090)+𝑥3=0 in .

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

Q6:

Determine the solution set of the equation loglog𝑥+𝑥+3=0 in .

  • A{27}
  • B127
  • C33
  • D133

Q7:

Find the solution set of loglog𝑥=(3𝑥+28) in .

  • A{4}
  • B{7}
  • C{11}
  • D{128}
  • E{64}

Q8:

Find the solution set of loglog𝑥49=1 in .

  • A{1,2}
  • B{14}
  • C7,149
  • D{1}

Q9:

Find the solution set of loglog𝑥=729 in .

  • A{81}
  • B{5}
  • C{9}
  • D{729}
  • E{14}

Q10:

Find the solution set of loglog𝑥+254=10 in .

  • A{5}
  • B{1,024}
  • C{4}
  • D{256}

Q11:

Find the solution set of loglog𝑥+252=10 in .

  • A{5}
  • B{3}
  • C{32}
  • D{2}
  • E{16}

Q12:

Solve loglog𝑥=64, where 𝑥.

  • A81
  • B27
  • C729
  • D127
  • E1729

Q13:

Solve loglog𝑥8𝑥=1, where 𝑥.

  • A{2,27}
  • B{2,3}
  • C{9,1}
  • D{8,8}
  • E{1,9}

Q14:

Solve loglog251+(𝑥+7)=8, where 𝑥.

Q15:

Solve loglog(7𝑥+194)=1, where 𝑥.

  • A𝑥=7
  • B𝑥=49
  • C𝑥=187
  • D𝑥=5

Q16:

Solve loglog𝑥+39=1, where 𝑥.

  • A𝑥=5 or 𝑥=5
  • B𝑥=34 or 𝑥=34
  • C𝑥=25 or 𝑥=25
  • D𝑥=60 or 𝑥=60
  • E𝑥=20 or 𝑥=20

Q17:

Solve logloglog36=0, where 𝑥.

Q18:

Find the solution set of 𝑥=10loglog in .

  • A{2}
  • B{64}
  • C{32}
  • D{6}

Q19:

The intensity levels 𝐼 of two earthquakes measured on a seismograph can be compared by the formula log𝐼𝐼=𝑀𝑀, where 𝑀 is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced another earthquake with a magnitude of 9.0. How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.

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