Worksheet: Solving Exponential Equations Using Logarithms

In this worksheet, we will practice using logarithms to solve exponential equations.

Q1:

Find, to the nearest hundredth, the value of ๐‘ฅ for which 2 = 9 ๏— ๏Šฐ ๏Šฎ .

Q2:

Use a calculator to find the value of ๐‘ฅ for which 9 = 8 ๏— . Give your answer correct to two decimal places.

Q3:

Solve 5 ๏€น 3 ๏… = 4 5 ๏Šง ๏Šฑ ๏— for ๐‘ฅ .

Q4:

Given that 2 = 1 8 ๏— and ๐‘› < ๐‘ฅ < ๐‘› + 1 , where ๐‘› is an integer, determine the value of ๐‘› .

Q5:

Find the value of ๐‘ฅ for which 8 = 1 2 . 7 ๏Šฉ ๏Šฑ ๏Šจ ๏— . Give your answer to the nearest tenth.

Q6:

Use a calculator to find the value of ๐‘ฅ for which 3 = 8 ๏Šฑ ๏Šช ๏— ๏Šฑ ๏Šฉ ๏— ๏Šฐ ๏Šช ๏Ž– ๏Šญ . Give your answer correct to two decimal places.

Q7:

Find all the values of ๐‘ฅ in โ„ which satisfy ๐‘ฅ = 1 0 ๏— ๏Šฐ ๏Šฌ ๏— ๏Šฐ ๏Šฌ .

  • A ๐‘ฅ = โˆ’ 6
  • B ๐‘ฅ = 1 0 or ๐‘ฅ = 6
  • C ๐‘ฅ = 1 0 or ๐‘ฅ = โˆ’ 6
  • D ๐‘ฅ = ยฑ 1 0
  • E ๐‘ฅ = ยฑ 1 0 or ๐‘ฅ = โˆ’ 6

Q8:

Determine the solution set of the equation 4 = 8 l o g l o g ๏— ๏Šช in โ„ .

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

Q9:

Given that 5 = 2 ๏— , determine the value of 2 5 ๏— .

Q10:

Find the solution set to the equation 1 5 = 6 4 ๏— . Give your answer to three decimal places.

  • A { 0 . 2 3 4 }
  • B { 1 . 5 3 6 }
  • C { 0 . 6 5 1 }
  • D { 4 . 2 6 7 }

Q11:

Find the solution set of 4 = ๐‘ฅ ๏— ๏Šฐ ๏Šฉ ๏— ๏Šฐ ๏Šฉ in โ„ .

  • A { 4 , 3 }
  • B { โˆ’ 4 , 3 }
  • C { 4 , โˆ’ 3 }
  • D { โˆ’ 4 , โˆ’ 3 }

Q12:

Given that 9 = 8 ๏Šช ๏Š ๏Šฑ ๏Šช ๏Šฎ ๏Šช ๏Š ๏Šฑ ๏Šช ๏Šฎ , where ๐‘› โˆˆ โ„ค , determine the value of ๐‘› .

Q13:

Find the solution set of 8 = 9 ๏— ๏Šฑ ๏Šญ ๏— ๏Šฑ ๏Šญ in โ„ .

  • A ๏ฌ 8 3 ๏ธ
  • B ๏ฌ 8 9 ๏ธ
  • C { 7 2 }
  • D { 7 }
  • E { โˆ’ 7 }

Q14:

Find the solution set of ๐‘ฅ = 1 0 l o g ๏‘ ๏Žฅ ๏— ๏Šฌ ๏Šช l o g in โ„ .

  • A { 6 }
  • B { 6 4 }
  • C { 2 }
  • D { 3 2 }

Q15:

Use a calculator to find the value of ๐‘ฅ for which 2 รท 5 = 2 ๏Šจ ๏Šฑ ๏Šซ ๏— ๏— ๏Šฐ ๏Šฉ . Give your answer correct to two decimal places.

Q16:

Use a calculator to find the value of ๐‘ฅ for which ๐‘ฅ = 9 7 . 4 ๏Šญ ๏Ž– ๏Šง . Give your answer correct to two decimal places.

Q17:

Find the value of ๐‘ฅ for which 3 6 = 2 ๏Šฉ ๏— . Give your answer to the nearest hundredth.

Q18:

Find the solution set of 8 + 8 = 6 5 ๏Šจ ๏Šฑ ๏— ๏Šฑ ๏— in โ„ .

  • A { 0 }
  • B { โˆ’ 1 }
  • C { 8 }
  • D { 1 }

Q19:

Solve 2 โ‹… 3 = 5 โ‹… 4 ๏— ๏— for ๐‘ฅ , giving your answer to three decimal places.

  • A ๐‘ฅ = โˆ’ 3 . 1 8 5
  • B ๐‘ฅ = 0 . 3 6 9
  • C ๐‘ฅ = 3 . 1 8 5
  • D ๐‘ฅ = โˆ’ 0 . 3 6 9
  • E ๐‘ฅ = 3 . 7 9 3

Q20:

The variables ๐‘Ž , ๐‘ , ๐‘› , and ๐‘Ÿ are related by the formula ๐‘ = ๐‘Ž ( ๐‘Ÿ โˆ’ 1 ) ๐‘Ÿ โˆ’ 1 ๏Š , where ๐‘Ž = 1 4 4 , ๐‘Ÿ = 3 2 , and ๐‘ = 1 . 8 9 9 ร— 1 0 ๏Šฉ . Find ๐‘› .

Q21:

What is the ๐‘ฅ -coordinate of the point where the graphs of ๐‘“ ( ๐‘ฅ ) = 9 ๏— ๏Šฑ ๏Šจ ๏Šฆ and ๐‘” ( ๐‘ฅ ) = 6 ๏— ๏Šฑ ๏Šจ ๏Šฆ intersect?

Q22:

Determine the solution set of ( ๐‘ฅ โˆ’ 2 ) = 1 ๏— ๏Šฑ ๏Šง ๏Šจ .

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

Q23:

If 6 = 9 ๏— , what is the value of 3 6 ๏— ?

Q24:

Find the solution set of 7 ร— 5 = 1 ๏Šญ ๏— ๏Šฑ ๏Šซ ๏Šง ๏Šซ ๏Šฑ ๏Šจ ๏Šง ๏— in โ„ .

  • A ๏ฌ โˆ’ 5 7 ๏ธ
  • B ๏ฌ 7 5 ๏ธ
  • C ๏ฌ โˆ’ 7 5 ๏ธ
  • D ๏ฌ 5 7 ๏ธ

Q25:

Find the solution set of 2 ร— ๐‘ฅ = 4 ๏— ๏Šฑ ๏Šฉ ๏Šซ ๏Šฑ ๏— .

  • A ๏ฌ 4 , 1 4 ๏ธ
  • B { 2 , โˆ’ 1 }
  • C { 4 , 3 }
  • D { 2 , 5 }
  • E { 2 , 4 }

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