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Worksheet: Solving Exponential Equations Using Logarithms

Q1:

Find, to the nearest hundredth, the value of π‘₯ for which 2 = 9 π‘₯ + 8 .

Q2:

Use a calculator to find the value of π‘₯ for which 9 = 8 π‘₯ . Give your answer correct to two decimal places.

Q3:

Solve 5 ο€Ή 3  = 4 5 1 βˆ’ π‘₯ for π‘₯ .

Q4:

Given that 2 = 1 8 π‘₯ and 𝑛 < π‘₯ < 𝑛 + 1 , where 𝑛 is an integer, determine the value of 𝑛 .

Q5:

Find the value of π‘₯ for which 8 = 1 2 . 7 3 βˆ’ 2 π‘₯ . Give your answer to the nearest tenth.

Q6:

Use a calculator to find the value of π‘₯ for which 3 = 8 βˆ’ 4 π‘₯ βˆ’ 3 π‘₯ + 4 . 7 . Give your answer correct to two decimal places.

Q7:

Find all the values of π‘₯ in ℝ which satisfy π‘₯ = 1 0 π‘₯ + 6 π‘₯ + 6 .

  • A π‘₯ = 1 0 or π‘₯ = 6
  • B π‘₯ = 1 0 or π‘₯ = βˆ’ 6
  • C π‘₯ = Β± 1 0
  • D π‘₯ = Β± 1 0 or π‘₯ = βˆ’ 6
  • E π‘₯ = βˆ’ 6

Q8:

Determine the solution set of the equation 4 = 8 l o g l o g π‘₯ 4 in ℝ .

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

Q9:

Given that 5 = 2 π‘₯ , determine the value of 2 5 π‘₯ .

Q10:

Find the solution set to the equation . Give your answer to three decimal places.

  • A
  • B
  • C
  • D

Q11:

Find the solution set of 4 = π‘₯ π‘₯ + 3 π‘₯ + 3 in ℝ .

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

Q12:

Given that 9 = 8 4 𝑛 βˆ’ 4 8 4 𝑛 βˆ’ 4 8 , where 𝑛 ∈ β„€ , determine the value of 𝑛 .

Q13:

Find the solution set of 8 = 9 π‘₯ βˆ’ 7 π‘₯ βˆ’ 7 in ℝ .

  • A { βˆ’ 7 }
  • B  8 9 
  • C { 7 2 }
  • D { 7 }
  • E  8 3 

Q14:

Find the solution set of π‘₯ = 1 0 l o g π‘₯ 6 π‘₯ 6 4 l o g in ℝ .

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

Q15:

Use a calculator to find the value of π‘₯ for which 2 Γ· 5 = 2 2 βˆ’ 5 π‘₯ π‘₯ + 3 . Give your answer correct to two decimal places.

Q16:

Use a calculator to find the value of π‘₯ for which π‘₯ = 9 7 . 4 7 . 1 . Give your answer correct to two decimal places.

Q17:

Find the value of π‘₯ for which 3 6 = 2 3 π‘₯ . Give your answer to the nearest hundredth.

Q18:

Find the solution set of 8 + 8 = 6 5 2 βˆ’ π‘₯ βˆ’ π‘₯ in ℝ .

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

Q19:

Solve 2 β‹… 3 = 5 β‹… 4 π‘₯ π‘₯ for π‘₯ , giving your answer to three decimal places.

  • A π‘₯ = 3 . 1 8 5
  • B π‘₯ = 3 . 7 9 3
  • C π‘₯ = 0 . 3 6 9
  • D π‘₯ = βˆ’ 3 . 1 8 5
  • E π‘₯ = βˆ’ 0 . 3 6 9

Q20:

The variables π‘Ž , 𝑐 , 𝑛 , π‘Ÿ a n d are related by the formula 𝑐 = π‘Ž ( π‘Ÿ βˆ’ 1 ) π‘Ÿ βˆ’ 1 𝑛 , where π‘Ž = 1 4 4 , π‘Ÿ = 3 2 , and 𝑐 = 1 . 8 9 9 Γ— 1 0 3 . Find 𝑛 .

Q21:

What is the π‘₯ -coordinate of the point where the graphs of 𝑓 ( π‘₯ ) = 9 π‘₯ βˆ’ 2 0 and 𝑔 ( π‘₯ ) = 6 π‘₯ βˆ’ 2 0 intersect?

Q22:

Determine the solution set of ( π‘₯ βˆ’ 2 ) = 1 π‘₯ βˆ’ 1 2 .

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

Q23:

If 6 = 9 π‘₯ , what is the value of 3 6 π‘₯ ?

Q24:

Find the solution set of 7 Γ— 5 = 1 7 π‘₯ βˆ’ 5 1 5 βˆ’ 2 1 π‘₯ in ℝ .

  • A  7 5 
  • B  βˆ’ 5 7 
  • C  βˆ’ 7 5 
  • D  5 7 

Q25:

Find the solution set of 6 Γ— π‘₯ = 7 7 7 6 π‘₯ βˆ’ 4 9 βˆ’ π‘₯ .

  • A  6 , 1 3 2 
  • B { 6 , 1 }
  • C  1 3 2 , 4 
  • D { 6 , 9 }
  • E  1 3 2 , 2 1 3 