e & ln (Edexcel International A Level Maths: Pure 3)

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"e"

What is e, the exponential function?

e Notes fig1, A Level & AS Maths: Pure revision notes

 

  • The exponential function is y = ex
    • e is an irrational number
    • e ≈ 2.718

  • As other exponential graphs do, y = ex
    • passes through (0, 1)
    • has the x-axis as an asymptote

What is the big deal with e?

e Notes fig2, A Level & AS Maths: Pure revision notes

 

e Notes fig3, A Level & AS Maths: Pure revision notes

 

  • y = ex has the particular property
    • dy/dx = ex
    • ie for every real number x, the gradient of y = ex is also equal to ex

      (see  Derivatives of Exponential Functions)

The negative exponential graph

e Notes fig4, A Level & AS Maths: Pure revision notes

 

  • y = e-x is a reflection in the y-axis of y = ex
    • They are of the form y = f(x) and y = f(-x)

      (see  Transformations of Functions - Reflections)

Exponential growth and decay

e Notes fig5, A Level & AS Maths: Pure revision notes

 

  • y = Aekx (k > 0) is exponential growth
  • y = Ae-kx (k > 0) is exponential decay
  • A is the initial value
  • k is a (usually positive) constant
  • “-“ is used in the equation making clear whether it is growth or decay

Worked example

e Example fig1, A Level & AS Maths: Pure revision notese Example fig2, A Level & AS Maths: Pure revision notes

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"ln"

What is ln? 

  • ln is a function that stands for natural logarithm
  • It is a logarithm where the base is the constant "e"
    • ln space x identical to log subscript straight e x
    • It is important to remember that ln is a function and not a number

What are the properties of ln? 

  • Using the definition of a logarithm you can see
    • ln space 1 equals 0
    • ln space straight e equals 1
    • ln space straight e to the power of x equals x 
    • ln space x is only defined for positive x
  • As ln is a logarithm you can use the laws of logarithms
    • ln space a plus ln space b equals ln left parenthesis a b right parenthesis
    • ln space a minus ln space b equals ln stretchy left parenthesis a over b stretchy right parenthesis
    • n space ln space a equals ln left parenthesis a to the power of n right parenthesis
  • Any logarithm can be written in terms of the natural logarithm using the change of base formula
    • log subscript a invisible function application b equals fraction numerator ln invisible function application b over denominator ln invisible function application a end fraction

How can I solve equations involving e & ln? 

  • The functions straight e to the power of x and ln space x are inverses of each other
    • If straight e to the power of straight f left parenthesis x right parenthesis end exponent equals straight g left parenthesis x right parenthesis then straight f stretchy left parenthesis x stretchy right parenthesis equals ln invisible function application space straight g stretchy left parenthesis x stretchy right parenthesis
    • If ln invisible function application space straight f left parenthesis x right parenthesis equals straight g open parentheses x close parentheses then straight f open parentheses x close parentheses equals straight e to the power of straight g open parentheses x close parentheses end exponent
  • If your equation involves "e" then try to get all the "e" terms on one side
    • If "e" terms are multiplied, you can add the powers
      • straight e to the power of x cross times straight e to the power of y equals straight e to the power of x plus y end exponent 
      • You can then apply ln to both sides of the equation
    • If "e" terms are added, try transforming the equation with a substitution
      • For example: If y equals straight e to the power of x then straight e to the power of 4 x end exponent equals y to the power of 4
      • You can then solve the resulting equation (usually a quadratic)
      • Once you solve for y then solve for x using the substitution formula
  • If your equation involves "ln", try to combine all "ln" terms together
    • Use the laws of logarithms to combine terms into a single term
    • If you have ln invisible function application space straight f open parentheses x close parentheses equals ln invisible function application space straight g left parenthesis x right parenthesis then solve straight f open parentheses x close parentheses equals straight g left parenthesis x right parenthesis
    • If you have ln invisible function application space straight f open parentheses x close parentheses equals k then solve straight f open parentheses x close parentheses equals straight e to the power of k

Worked example

3-1-1-ln-we-solution

Examiner Tip

  • Always simplify your answer if you can
    • for example, 1 half ln space 25 space equals space ln space square root of 25 equals ln space 5
    • you wouldn't leave your final answer as square root of 25 so don't leave your final answer as 1 half ln space 25

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Dan

Author: Dan

Expertise: Maths

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.