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IBMaths: AI HLDPRevision Notes1. Number & Algebra1.2 Exponentials & Logs1.2.2 Logarithms

Logarithms (DP IB Maths: AI HL)

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Introduction to Logarithms

What are logarithms?

  • A logarithm is the inverse of an exponent
    • If a to the power of x equals b then log subscript a open parentheses b close parentheses equals x where a > 0, b > 0, a ≠ 1
      • This is in the formula booklet
      • The number a is called the base of the logarithm
      • Your GDC will be able to use this function to solve equations involving exponents
  • Try to get used to ‘reading’ logarithm statements to yourself
    • log subscript a left parenthesis b right parenthesis space equals space x would be read as “the power that you raise a to, to get b, is x”
    • So log subscript 5 125 space equals space 3 would be read as “the power that you raise 5 to, to get 125, is 3”
  • Two important cases are:
    • ln space x equals log subscript straight e open parentheses x close parentheses
      • Where e is the mathematical constant 2.718…
      • This is called the natural logarithm and will have its own button on your GDC
    • log space x equals log subscript 10 open parentheses x close parentheses
      • Logarithms of base 10 are used often and so abbreviated to log x

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 GDCs to find the value of log subscript 2 10

Examiner Tip

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

Worked example

Solve the following equations:

i)
x equals log subscript 3 27,
 

ai-sl-1-1-2intro-to-logs-we-i

ii)
2 to the power of x equals 21.4, giving your answer to 3 s.f.
 
ai-sl-1-1-2intro-to-logs-we-ii

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Laws of Logarithms

What are the laws of logarithms?

  • Laws of logarithms allow you to simplify and manipulate expressions involving logarithms
    • The laws of logarithms are equivalent to the laws of indices
  • The laws you need to know are, given  a comma space x comma space y space greater than space 0:
    • log subscript a x y equals blank log subscript a x plus blank log subscript a y
      • This relates to a to the power of x cross times blank a to the power of y equals a to the power of x plus y end exponent
    • log subscript a x over y equals blank log subscript a x blank negative space log subscript a y 
      • This relates to a to the power of x divided by blank a to the power of y equals a to the power of x minus y end exponent
    • log subscript a x to the power of m equals blank m log subscript a x 
      • This relates to left parenthesis a to the power of x right parenthesis to the power of y equals a to the power of x y end exponent
  • These laws are in the formula booklet so you do not need to remember them
    • You must make sure you know how to use them

Laws of Logarithms Notes fig2

Useful results from the laws of logarithms

  • Given a space greater than space 0 space comma space a space not equal to space 1
    • log subscript a 1 equals blank 0
      • This is equivalent to a to the power of 0 equals 1
  • If we substitute b for a into the given identity in the formula booklet
    • a to the power of x equals b space left right double arrow space log subscript a b space equals space x where a space greater than space 0 comma space b space greater than space 0 comma space a space not equal to space 1
    • a to the power of x space equals space a space left right double arrow space log subscript a a space equals space x gives a to the power of 1 space equals space a space left right double arrow space log subscript a a space equals space 1 
      • This is an important and useful result
  • Substituting this into the third law gives the result
    • log subscript a a to the power of k equals blank k
  • Taking the inverse of its operation gives the result
    • a to the power of log subscript a x end exponent equals blank x
  • From the third law we can also conclude that
    • log subscript a 1 over x equals blank minus log subscript a x

Laws of Logarithms Notes fig3

  • These useful results are not in the formula booklet but can be deduced from the laws that are
  • Beware…
    • …log subscript a open parentheses x plus y close parentheses space not equal to space log subscript a x plus log subscript a y
  • These results apply to ln space x space left parenthesis log subscript e x right parenthesis too
    • Two particularly useful results are
      • ln space e to the power of x space equals space x
      • e to the power of ln x end exponent space equals space x
  • Laws of logarithms can be used to …
    • simplify expressions
    • solve logarithmic equations
    • solve exponential equations

Examiner Tip

  • Remember to check whether your solutions are valid
    • log (x+k) is only defined if x > -k
    • You will lose marks if you forget to reject invalid solutions

Worked example

a)

Write the expression 2 space log space 4 space minus space log space 2 in the form log space k, where k space element of space straight integer numbers.

 aa-sl-1-2-2-laws-of-logs-we-solution-part-a

b)   Hence, or otherwise, solve 2 space log space 4 minus log space 2 equals negative log blank 1 over x.

aa-sl-1-2-2-laws-of-logs-we-solution-part-b

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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.