Logarithms (Cambridge (CIE) IGCSE International Maths)

Logarithms

What are logarithms?

• A logarithm is the inverse function of an exponential function

  • Making the subject of gives

    • This is pronounced "log base a of b"

    • This is the definition of a logarithm

    • The number is called the base of the logarithm

• If no base is given, it means the base number is 10

  • means

    • In the past, this was a common base to use

    • The log button on a calculator is always base 10

How do I interpret logarithms?

• log base a of b, , is the power that you raise to, to get

  • This is the power required to take the small number, , to the big number,

  • For example, is the power needed to get 5 to 125

    • The power needed is 3 (because 5³ = 125)

    • So

  • Similarly

    • 

    • 

    • (as )

  • It works for negative powers (reciprocals)

    • (as )

    • 

  • It works for fractional powers (roots)

    • (as )

    • 

• Where possible, use a calculator to check

  • Use the button which allows you to change the base number

    • This is a different button to the log button (base 10)

How do I convert between logarithms and exponentials?

• and are mathematically identical statements

  • You can convert between the two

• Learn to read as " is the power that you raise to, to get "

  • This allows you to convert logs to exponentials

    • 

• This also works the other way

  • means

    • Though it is often easier to think of making the subject

    • To make the subject, take log base a of both sides,

• This means you can relate logarithms to exponential graphs

  • e.g. the graph of passes through the point

    • so

How do I solve equations with logarithms?

• Equations where is in the power are called exponential equations

  • Some can be solved easily by inspection (seeing the answer)

    • For example, is solved by

  • Others cannot

    • For example,

  • These must be solved using logarithms by making the subject

    • means

    • Type this into your calculator to get

• Alternatively, you can also solve equations using logs of base 10 only

  • The rule for base 10 is if then where means base 10

  • For example, means

How do I use logarithms in compound interest questions?

• You can use logarithms to work out the power, , in compound interest questions

  • e.g.you put £500 into a bank account with 5% interest per year; after how many whole years does it exceed £900?

    • 

    • Make the subject: so (or )

    • This gives so years

    • (it has not exceeded 900 after 12 years)

How do I find inverse functions using logarithms?

• For exponential functions, you can use logarithms to find their inverse functions

  • Use the standard process for finding inverse functions

  • e.g.

    • then swap and to get

    • Make the subject using logarithms:

    • Finally make the subject to give