Laws of Logarithms (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Laws of logarithms

What are the laws of logarithms?

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

  • The laws of logarithms are equivalent to the laws of indices

• The laws you need to know are, given $a,x,y>0$:

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

    • This relates to $a^x\times { a}^y=a^{x+y}$

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

    • This relates to $a^x÷ a^y=a^{x-y}$

  • ${\log }_a{x}^m= m{\log }_{a}x$ 

    • This relates to $(a^x{)}^y=a^{xy}$

• There are also some particular results these lead to

  • ${\log }_{a}a=1$

  • ${\log }_a{a}^x=x$

  • $a^{{\log }_{a}x}=x$

  • ${\log }_{a}1=0$

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

• Beware…

  • …${\log }_a\left(x+y\right)\ne {\log }_{a}x+{\log }_{a}y$

• These results apply to $\ln x\left({\log }_{e}x\right)$ too

  • Two particularly useful results are

    • $\ln e^x=x$

    • $e^{\ln x}=x$

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

How do I change the base of a logarithm?

• The formula for changing the base of a logarithm is

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

• The value you choose for b does not matter, however if you do not have a calculator, you can choose b such that the problem will be possible to solve

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 of the logarithm and then apply the laws of logarithms

• Changing the base of a logarithm can be particularly useful if you need to evaluate a log problem without a calculator

  • Choose the base such that you would know how to solve the problem from the equivalent exponent

• 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 can be useful when evaluating 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 ${\log }_{x}x=1$