Laws of Logarithms (Cambridge (CIE) A Level Maths: Pure 3): 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

• Results apply to ln too

  • $\ln x\equiv {\log }_{\mathrm{e}}x$

  • In particular ${\mathrm{e}}^{\ln x}=x$ and $\ln \left({\mathrm{e}}^x\right)=x$

How do I use log laws?

• Laws of logarithms can be used to …

  • … simplify expressions

  • … solve logarithmic equations

  • … solve exponential equations