Properties of Logarithms (Edexcel IGCSE Further Pure Maths): Revision Note

Laws of Logarithms

What are the laws of logarithms?

• Laws of logarithms allow you to simplify and manipulate expressions involving logarithms

  • They can help with solving exponential and logarithmic equations

  • The laws of logarithms are closely related to the laws of indices

• You need to know the following laws, which are valid for $\mathit{a}\mathbf{,}\mathit{x}\mathbf{,}\mathit{y}\mathbf{>}\mathbf{0}$,  $\mathit{a}\mathbf{\ne }\mathbf{1}$:

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

    • "log of a product is equal to the sum of the logs"

    • This relates to $a^m\times { a}^n=a^{m+n}$

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

    • "log of a division is equal to the difference of the logs"

    • This relates to $a^m÷ a^n=a^{m-n}$

  • ${\log }_a{x}^k= k{\log }_{a}x$ 

    • "power in a log may be brought down as a multiplier in front of the log"

    • note that  ${\log }_a{x}^k={\log }_a\left(x^k\right)$

    • This relates to $(a^m{)}^n=a^{mn}$

  • ${\log }_{a}a=1$

    • This relates to  $a^1=a$

  • ${\log }_{a}1=0$

    • This relates to  $a^{0=1}$

• Be careful

  • With the first two laws the logs on the right-hand side must have the same base

    • Logs with different bases cannot be combined using those laws

  • Also note the following (students often make these mistakes on the exam):

    • ${\log }_a\left(x+y\right)$ is not equal to ${\log }_{a}x+{\log }_{a}y$

    • ${\log }_a\left(x-y\right)$ is not equal to ${\log }_{a}x-{\log }_{a}y$

What are some other useful properties of logarithms?

• You should also be familiar with the following properties of logarithms

  • ${\log }_a{a}^k= k$

    • This can be derived from the third and fourth laws above

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

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

    • Then use the third law above

  • ${\log }_a\left(a^x\right)=a^{{\log }_{a}x}= x$

    • Because ${\log }_{a}x$ and $a^x$ are inverse functions

    • "log cancels exponential and exponential cancels log"

• Also remember that $\ln x$ is another way of writing ${\log }_{\mathrm{e}}x$

  • All the laws and properties apply to $\ln x$ as well

  • This includes  $\ln \left({\mathrm{e}}^x\right)={\mathrm{e}}^{\ln x}\mathit{=}x$

Change of Base

Why change the base of a logarithm?

• The laws of logarithms can only be used if the logs have the same base

  • If a problem involves logarithms with different bases

    • you can change the base(s) of the logarithm(s)

    • and then apply the logarithm laws

• Changing the base can also allow a log problem to be solved without a calculator

  • Choose a base that allows you to solve the problem using the equivalent exponent

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 is on the formula sheet in the exam paper

  • So you don't need to remember it

  • But you do need to be able to use it

• The change of base formula also leads to the following useful result:

  • ${\log }_{a}b=\frac{1}{{\log }_{b}a}$

    • This is not on the formula sheet

    • But it can be derived from the change of base formula:

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