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Worksheet: Converting Exponential Expressions to Logarithmic Form

In this worksheet, we will practice utilizing the relationship between logarithms and exponentials to convert an exponential expression to its logarithmic form.

Q1:

Express 4 = 1 0 in logarithmic form.

  • A l o g 0 4 = 1
  • B l o g 4 0 = 1
  • C l o g 1 4 = 0
  • D l o g 4 1 = 0

Q2:

Express 9 7 = 8 1 4 9 2 in its equivalent logarithmic form.

  • A l o g 8 1 4 9 = 1 8 7
  • B l o g 2 8 1 4 9 = 9 7
  • C l o g 8 1 4 9 9 7 = 2
  • D l o g 9 7 8 1 4 9 = 2

Q3:

Express 3 4 3 8 = 7 2 1 3 in its equivalent logarithmic form.

  • A l o g 7 2 = 1 0 2 9 8
  • B l o g 1 3 7 2 = 3 4 3 8
  • C l o g 7 2 3 4 3 8 = 1 3
  • D l o g 3 4 3 8 7 2 = 1 3

Q4:

Express 2 4 3 = 3 1 5 in its equivalent logarithmic form.

  • A 2 4 3 √ 3 = 1 5
  • B l o g 1 5 3 = 2 4 3
  • C l o g 3 = 2 4 3 5
  • D l o g 2 4 3 3 = 1 5
  • E l o g 3 1 5 = 2 4 3

Q5:

Express 4 = 1 1 6 − 2 in its equivalent logarithmic form.

  • A − 2 1 1 6 = 4
  • B l o g − 2 1 1 6 = 4
  • C l o g 1 1 6 = − 8
  • D l o g 4 1 1 6 = − 2
  • E l o g 1 1 6 4 = − 2

Q6:

Express ( 0 . 7 ) = 0 . 3 4 3 3 in its equivalent logarithmic form.

  • A 0 . 7 √ 0 . 3 4 3 = 3
  • B l o g 3 0 . 3 4 3 = 0 . 7
  • C l o g 0 . 3 4 3 = 2 . 1
  • D l o g 0 . 7 0 . 3 4 3 = 3
  • E l o g 0 . 3 4 3 0 . 7 = 3

Q7:

Express ( 0 . 1 6 ) = 0 . 4 1 2 in its equivalent logarithmic form.

  • A 0 . 4 √ 0 . 1 6 = 1 2
  • B l o g 0 . 4 0 . 1 6 = 1 2
  • C l o g 0 . 1 6 = 0 . 8
  • D l o g 0 . 1 6 0 . 4 = 1 2
  • E l o g 1 2 0 . 4 = 0 . 1 6

Q8:

Express 2 = 4 √ 2 5 2 in its equivalent logarithmic form.

  • A l o g 4 √ 2 = 5
  • B l o g 5 2 4 √ 2 = 2
  • C l o g 4 √ 2 2 = 5 2
  • D l o g 2 4 √ 2 = 5 2
  • E l o g 4 √ 2 = 5 2 2

Q9:

Express 1 0 = 1 0 0 0 3 in its equivalent logarithmic form.

  • A − 3 √ 1 0 0 0 = 1 0
  • B l o g 3 = 1 0 0 0
  • C l o g 1 0 0 0 = 3 0
  • D l o g 1 0 0 0 = 3
  • E l o g 3 1 0 0 0 = 1 0

Q10:

Express 2 = 1 1 6 √ 2 − 9 2 in its equivalent logarithmic form.

  • A l o g 1 1 6 √ 2 = − 9
  • B l o g − 9 2 1 1 6 √ 2 = 2
  • C l o g 1 1 6 √ 2 2 = − 9 2
  • D l o g 2 1 1 6 √ 2 = − 9 2
  • E l o g 9 2 1 1 6 √ 2 = 2

Q11:

Express √ 2 = 5 1 2 1 8 in its equivalent logarithmic form.

  • A l o g 5 1 2 = 1 8 √ 2
  • B l o g 1 8 5 1 2 = √ 2
  • C l o g 5 1 2 √ 2 = 1 8
  • D l o g √ 2 5 1 2 = 1 8

Q12:

Solve for 𝑥 2 = 3 : .

  • A 𝑥 = 3 3 − 2 2 − 2 − 3 3 l o g l o g l o g l o g
  • B 𝑥 = 2 3 − 2 3 − 3 2 l o g l o g l o g l o g
  • C 𝑥 = 2 + 2 3 3 − 2 2 l o g l o g l o g l o g
  • D 𝑥 = 3 + 3 2 2 3 − 2 l o g l o g l o g l o g

Q13:

Given that 2 − 6 = 1 0 4 𝑥 , find the value of 𝑥 .

Q14:

The value of an antique painting increases every year. The painting is currently worth $ 1 2 0 0 0 , and it is expected to be worth $ 2 0 0 0 0 in 8 years’ time.

Write an equation that can be used to find 𝑟 , the rate at which its value increases.

  • A 1 2 0 0 0 ( 1 + 𝑟 ) = 2 0 0 0 0 8
  • B 1 2 0 0 0 ( 𝑟 ) = 2 0 0 0 0 8
  • C 1 2 0 0 0 𝑟 1 0 0 = 2 0 0 0 0 8
  • D 1 2 0 0 0 1 + 𝑟 1 0 0 = 2 0 0 0 0 8
  • E 1 2 0 0 0 1 − 𝑟 1 0 0 = 2 0 0 0 0 8

Q15:

Solve for 𝑥 2 = 1 2 : 𝑥 + 5 .

  • A 𝑥 = 2 1 2 − 5 l o g l o g
  • B 𝑥 = 6 − 5 l o g
  • C 𝑥 = 1 6 − 5 l o g
  • D 𝑥 = 1 2 2 − 5 l o g l o g

Q16:

The population of a rare orchid declines by 𝑟 % every year. There are currently only 99 of these orchids left, and conservationists predict that only 50 will be left in 5 years. Write an equation that can be used to find 𝑟 , the rate of decline.

  • A 9 9 ( 1 − 𝑟 ) = 5 0 5
  • B 9 9 ( 𝑟 ) = 5 0 5
  • C 9 9 𝑟 1 0 0 = 5 0 5
  • D 9 9 1 − 𝑟 1 0 0 = 5 0 5
  • E 9 9 1 + 𝑟 1 0 0 = 5 0 5

Q17:

Solve 1 8 = 6 4 𝑥 .

  • A 𝑥 = 3
  • B 𝑥 = 2
  • C 𝑥 = − 3
  • D 𝑥 = − 2
  • E 𝑥 = 6

Q18:

Find the solution set of l o g l o g 2 2 𝑥 𝑦 = 5 − 4 and l o g l o g 2 1 0 2 𝑦 = 1 0 − 1 in ℝ × ℝ .

  • A { ( 2 , 4 ) }
  • B { ( 2 , 8 ) }
  • C { ( 8 , 1 ) }
  • D { ( 4 , 2 ) }

Q19:

Given that 4 = 3 𝑥 and 3 = 6 4 𝑦 , determine the value of 𝑥 𝑦 .

Q20:

Find the solution set of 9 = 1 7 2 9 𝑥 − 4 0 𝑥 2 in ℝ .

  • A { 5 , 8 }
  • B { − 5 , 8 }
  • C { − 5 , − 8 }
  • D { 5 , − 8 }

Q21:

Given that 7 6 × 𝑥 = 7 6 2 8 3 5 , find 𝑥 .

  • A 7 6
  • B 7 6 6 3
  • C 7 6 3 6
  • D 7 6 7

Q22:

Which of the following satisfies the equation 6 × 6 × 3 × 3 × 3 × 3 = 9 × 𝑥 ?

  • A 6 + 3 2 2
  • B 1 8 3
  • C 6 2
  • D 1 8 2
  • E 3 2

Q23:

Determine the solution set of the equations 𝑥 = 7 𝑥 + 6 𝑦 and l o g 𝑥 2 7 = 𝑦 , given that it is a subset of ℝ × ℝ .

  • A { ( 6 , 3 ) }
  • B { ( 3 , 7 ) }
  • C { ( 6 , 7 ) }
  • D { ( 3 , 3 ) }
  • E { ( 4 , 3 ) }

Q24:

Given that 𝑓 ( 𝑥 ) = 2 𝑥 , determine the solution set of 𝑓 ( 𝑥 ) − 2 4 𝑓 𝑥 2 = − 1 2 8 .

  • A { 5 , − 5 }
  • B { 1 6 , 3 2 }
  • C { 8 , − 8 }
  • D { 6 , 8 }
  • E { − 1 , 1 }

Q25:

Find the solution set of l o g l o g l o g l o g 2 2 2 2 𝑥 + 𝑦 + 3 2 = 5 + 4 8 and 𝑥 + 𝑦 = 1 4 in ℝ × ℝ .

  • A { ( 6 , 1 6 ) , ( 1 6 , 6 ) }
  • B { ( 5 , 9 ) , ( 9 , 5 ) }
  • C { ( 2 , 1 2 ) , ( 1 2 , 2 ) }
  • D { ( 6 , 8 ) , ( 8 , 6 ) }
  • E { ( 3 0 , 8 ) , ( 4 0 , 6 ) }

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