Laws of Logarithms (Cambridge International A-Level Mathematics)

Laws of logarithms

What are the laws of logarithms?

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

  • a^mx aⁿ= a^m+n
  • (a^m)ⁿ= a^mn

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

  • $\mathrm{log}_{a}xy=\mathrm{log}_{a}x+\mathrm{log}_{a}y$
  • $\mathrm{log}_{a}(\frac{x}{y})=\mathrm{log}_{a}x-\mathrm{log}_{a}y$
  • $\mathrm{log}_{a}x^{k}=k\mathrm{log}_{a}x$

• There are also some particular results these lead to

  • $\mathrm{log}_{a}a=1$
  • $\mathrm{log}_{a}a^{x}=x$
  • $a^{\mathrm{log}_{a}x}=x$
  • $\mathrm{log}_{a}1=0$
  • $\mathrm{log}_{a}(\frac{1}{x})=-\mathrm{log}_{a}x$

• Two of these were seen in the notes Logarithmic Functions

• Beware …

  • … log (x + y) ≠ log x + log y

• Results apply to ln too

  • $\mathrm{ln}x\equiv\mathrm{log}_{e}x$
  • In particular $e^{\mathrm{ln}x}=x$ and $\mathrm{ln}(e^{x})=x$

How do I use log laws?

• Laws of logarithms can be used to …

  • … simplify expressions
  • … solve logarithmic equations
  • … solve exponential equations

Exam Hint

• 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