Laws & Change of Base (Edexcel International A Level (IAL) Maths: Pure 2): Revision Note

Laws of logarithms

What are the laws of logarithms?

• There are many laws or rules of indices, for example

  • a^m x aⁿ = a^m+n

  • (a^m)ⁿ = a^mn

• There are equivalent laws of logarithms (for a > 0)

  • ${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$

  • ${\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y$

  • ${\log }_a{x}^k=k{\log }_{a}x$ 

• There are also some particular results these lead to

  • ${\log }_{a}a=1$

  • ${\log }_a{a}^x=x$

  • $a^{{\log }_{a}x}=x$

  • ${\log }_{a}1=0$

  • ${\log }_a\left(\frac{1}{x}\right)=-{\log }_{a}x$

• Two of these were seen in the notes Logarithmic Functions

• Beware …

  • … log (x + y) ≠ log x + log y

How do I use the laws of logarithms?

• Laws of logarithms can be used to …

  • … simplify expressions

  • … solve logarithmic equations

  • … solve exponential equations

Change of base

What is the change of base formula? 

• We can rewrite a logarithm as a multiple of a logarithm of any different positive base using the formula

${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$

How do I use the change of base formula? 

• This formula had more use when calculators were less advanced

  • Some old calculators only had a button for logarithm of base 10

  • To calculate ${\log }_{5}7$on these calculators you would have to enter

    • $\frac{{\log }_{10}7}{{\log }_{10}5}$

• The formula is only needed in a small number of cases

  • This is given in the formulae booklet in case it is needed

• The formula can be useful when evaluation a logarithm where the two numbers are powers of a common number

  • ${\log }_{4}8=\frac{{\log }_{2}8}{{\log }_{2}4}=\frac{3}{2}$

• The formula can be useful when you are solving equations and two logarithms have different bases

  • For example, if you have ${\log }_{3}k$ and ${\log }_{9}n$ within the same equation

    • You can rewrite ${\log }_{9}n$ as $\frac{{\log }_{3}n}{{\log }_{3}9}$ which simplifies to $\frac{1}{2}{\log }_{3}n$

    • Or you can rewrite ${\log }_{3}k$ as $\frac{{\log }_{9}k}{{\log }_{9}3}$ which simplifies to $2{\log }_{9}k$

• The formula also allows you to derive and use a formula for switching the numbers:

${\log }_{a}x=\frac{1}{{\log }_{x}a}$

• Using the fact that ${\log }_{x}x=1$