Logarithmic Functions (Cambridge O Level Additional Maths)

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Logarithmic Functions

What are logarithmic functions?

  • A logarithm is the inverse of raising to a power 
  • If a space equals space b to the power of x then log subscript b a space equals space x
    • a space greater than space 0
    • b is called the base of the logarithm
  • Try to get used to ‘reading’ logarithm statements to yourself
    •  log subscript... end subscript open parentheses... close parentheses space equals would be read as “the power that you raise ... to, to get ..., is 
    • So log subscript 5 open parentheses 125 close parentheses equals 3 would be read as “the power that you raise 5 to, to get 125, is 3”

Connection between exponentials and logarithms

  • A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functions

Logarithms as functions with their inverses

Why use logarithms?

  • Logarithms allow us to solve equations where the exponent is the unknown value
    • We can solve some of these by inspection
      • For example, for the equation 2x = 8 we know that x must be 3
    • Logarithms allow use to solve more complicated problems
      • For example, the equation 2x = 10 does not have a clear answer
      • Instead, we can use our calculator to find the value of log subscript 2 10

How do I use logarithms?

Finding the values of logarithms

  •  Recognising the rules of logarithms allows expressions to be simplified

Working out a complicated logarithm

  • Recognition of common powers helps in simple cases
    • Powers of 2: 20 = 1, 21 = 2, 22 = 4, 23 = 8, 24 =16, …
    • Powers of 3: 30 = 1, 31 = 3, 32 = 9, 33 = 27, 34 = 81, …
    • The first few powers of 4, 5 and 10 should also be familiar
    For more awkward cases a calculator is needed Using a calculator to find logarithms
  • Calculators can have, possibly, three different logarithm buttons

logarithm button with a base 

  • This button allows you to type in any number for the base

The button for ln

  • Natural logarithms (see “e”)

The button for log (base 10)

  • Shortcut for base 10 although SHIFT button needed 
  • Before calculators, logarithmic values had to be looked up in printed tables

What notation might I see with logarithms?

Abbreviations for common logarithms

  • 10 is a common base
    • log10 x is abbreviated to log x or lg x
  • The value e is another common base
    • loge x is abbreviated to ln x
  • (log x)2 ≠ log x2

Examiner Tip

  • Before going into the exam, make sure you are completely familiar with your calculator and know how to use its logarithm functions

ln x

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
  • The natural logarithm (ln space x) and the exponential function (straight e to the power of x) are inverses of each other 
  • It is defined for all positive numbers (x space greater than space 0)
    • ln space x cannot be defined for negative numbers or x space equals space 0

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

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 x equals a then x equals ln invisible function application space a
    • If ln invisible function application space x equals a then x equals straight e to the power of a
    • 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

Examiner Tip

  • If you're working on the non-calculator exam paper you may need to leave answers as exact values so using lnis a good idea
    • This can be solved with ewhich can then be left as an exact value

Worked example

Solve the equation straight e to the power of 2 x space end exponent equals space 5, leaving your answer as an exact value. 

Take the natural logarithm of both sides. 

ln space open parentheses straight e to the power of 2 x end exponent close parentheses space equals space ln space 5

Use the property that ln space open parentheses straight e to the power of a close parentheses space equals space a.

2 x space equals space ln space 5 

Divide both sides by 2. 

table row cell x space end cell equals cell space fraction numerator ln space 5 over denominator 2 end fraction end cell end table

Do not use your calculator to evaluate this as the question asks for the answer given as an exact value.

bold italic x bold space bold equals bold space fraction numerator bold ln bold space bold 5 over denominator bold 2 end fraction

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.