Worksheet: Base e Exponential Equations

In this worksheet, we will practice solving exponential equations with base e by using natural logarithms knowing that ln(e) = 1.

Q1:

Solve 3 𝑒 4 = 8 𝑥 , giving your answer correct to three decimal places.

  • A 𝑥 = 2 . 4 8 5
  • B 𝑥 = 0 . 2 8 8
  • C 𝑥 = 1 . 9 4 6
  • D 𝑥 = 1 . 3 8 6
  • E 𝑥 = 0 . 9 8 1

Q2:

Find, to the nearest thousandth, the value of 𝑥 such that 𝑒 = 1 9 4 𝑥 3 .

Q3:

Solve 2 = 5 𝑒 𝑥 𝑥 1 for 𝑥 , giving your answer to three decimal places.

  • A 𝑥 = 1 . 4 6 4
  • B 𝑥 = 1 . 9 8 6
  • C 𝑥 = 1 . 7 5 6
  • D 𝑥 = 1 . 9 8 6
  • E 𝑥 = 1 . 4 6 4

Q4:

Consider the function 𝑓 ( 𝑥 ) = 4 𝑒 1 2 𝑒 + 1 .

How many solutions does the equation 𝑓 ( 𝑥 ) = 2 have?

  • A2
  • B1
  • C0

How many solutions does the equation 𝑓 ( 𝑥 ) = 1 have?

  • A1
  • B2
  • C0

How many solutions does the equation 𝑓 ( 𝑥 ) = 4 have?

  • A1
  • B2
  • C0

What is the range of the function 𝑓 ( 𝑥 ) ?

  • A 1 < 𝑦 < 2
  • B 2 . 5 < 𝑦 < 3
  • C 2 < 𝑦 <
  • D < 𝑦 < 2
  • E 0 < 𝑦 < 2

Q5:

Consider the function 𝑓 ( 𝑥 ) = 4 𝑒 1 3 𝑒 + 1 .

How many solutions does the equation 𝑓 ( 𝑥 ) = 3 2 have?

  • A2
  • B1
  • C0

What is the range of the function 𝑓 ( 𝑥 ) ?

  • A 1 < 𝑦 < 4 3
  • B 0 < 𝑦 < 4
  • C 1 < 𝑦 < 4 3
  • D 1 . 5 < 𝑦 <
  • E 1 < 𝑦 < 3

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