Laws of Logarithms (DP IB Analysis & Approaches (AA): HL): Revision Note

Laws of logarithms

What are the laws of logarithms?

• The laws of logarithms (log laws) you need to know are:

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

  • ${\log }_a\frac{x}{y}= {\log }_{a}x {-\log }_{a}y$

  • ${\log }_a{x}^m= m{\log }_{a}x$

• These hold for $a,x,y>0$

Useful results from the laws of logarithms

• ${\log }_{a}1=0$

  • This is equivalent to $a^0=1$

• ${\log }_{a}a=1$

  • This is equivalent to $a^1=a$

• ${\log }_a{a}^k= k$

  • because ${\log }_a{a}^k= k{\log }_{a}a=k\times 1$

• $a^{{\log }_{a}x}= x$

  • because logarithms and powers are inverses

• ${\log }_a\frac{1}{x}=-{\log }_{a}x$

  • because ${\log }_a\frac{1}{x}={\log }_{a}1-{\log }_{a}x=0-{\log }_{a}x$

• The useful results can be applied to $\ln x\left({\log }_{e}x\right)$ too

  • Two particularly useful results are

    • $\ln {\mathrm{e}}^x=x$

    • ${\mathrm{e}}^{\ln x}=x$

When are logarithms undefined?

• You cannot take the log of zero or the log of a negative number

  • ${\log }_{a}x$ is defined for $x>0$

  • ${\log }_{a}x$ is undefined for $x\le 0$

• Similarly

  • ${\log }_a\left(x-5\right)$ is defined for $x>5$

  • ${\log }_a\left(x-5\right)$ is undefined for $x\le 5$

  • etc

Change of base

How do I change the base of a logarithm?

• The formula for changing the base of a logarithm is

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

• This allows you to change logarithms into a more useful base

  • which is helpful in non-calculator questions