Laws & Change of Base (Edexcel IAL Maths)

Laws of Logarithms

Spec point — 5.2

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

How do I use the laws of logarithms?

• 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

Change of Base

Spec point — 5.3

What is the change of base formula?

• We can rewrite a logarithm as a multiple of a logarithm of any different positive base using the formula

$log_{a} x = \frac{log_{b} x}{log_{b} a}$

How do I use the change of base formula?

• This formula had more use when calculators were less advanced

  • Some old calculators only had a button for logarithm of base 10

  • To calculate $\mathrm{log}_{5}7$on these calculators you would have to enter

    • $\frac{log_{10} 7}{log_{10} 5}$

• The formula is only needed in a small number of cases

  • This is given in the formulae booklet in case it is needed

• The formula can be useful when evaluation a logarithm where the two numbers are powers of a common number

  • $\mathrm{log}_{4}8=\frac{\mathrm{log}_{2}8}{\mathrm{log}_{2}4}=\frac{3}{2}$

• The formula can be useful when you are solving equations and two logarithms have different bases

  • For example, if you have $\mathrm{log}_{3}k$ and $\mathrm{log}_{9}n$ within the same equation

    • You can rewrite $\mathrm{log}_{9}n$ as $\frac{log_{3} n}{log_{3} 9}$ which simplifies to $\frac{1}{2}\mathrm{log}_{3}n$
    • Or you can rewrite $\mathrm{log}_{3}k$ as $\frac{log_{9} k}{log_{9} 3}$ which simplifies to $2\mathrm{log}_{9}k$

• 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 $\mathrm{log}_{x}x=1$

Exam Hint

• It is very rare that you will need to use the change of base formula
• Only use it when the bases of the logarithms are different