Logarithmic Functions (Cambridge (CIE) IGCSE Additional Maths): Revision Note

Logarithmic functions

What are logarithmic functions?

• A logarithm is the inverse of raising to a power 

• If $a=b^x$ then ${\log }_{b}a=x$

  • $a>0$

  • $b$ is called the base of the logarithm

• Try to get used to ‘reading’ logarithm statements to yourself

  •  ${\log }_{...}\left(...\right)=$ would be read as “the power that you raise ... to, to get ..., is ”

  • So ${\log }_5\left(125\right)=3$ would be read as “the power that you raise 5 to, to get 125, is 3”

• A logarithm is the inverse of raising to a power so we can use rules to simplify logarithmic functions

Why use logarithms?

• Logarithms allow us to solve equations where the exponent is the unknown value

  • We can solve some of these by inspection

    • For example, for the equation 2^x = 8 we know that x must be 3

  • Logarithms allow use to solve more complicated problems

    • For example, the equation 2^x = 10 does not have a clear answer

    • Instead, we can use our calculator to find the value of ${\log }_{2}10$

How do I use logarithms?

•  Recognising the rules of logarithms allows expressions to be simplified

• Recognition of common powers helps in simple cases

  • Powers of 2: 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ =16, …

  • Powers of 3: 3⁰ = 1, 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81, …

  • The first few powers of 4, 5 and 10 should also be familiar

  For more awkward cases a calculator is needed 

  

• Calculators can have, possibly, three different logarithm buttons

• This button allows you to type in any number for the base

• Natural logarithms (see “e”)

• Shortcut for base 10 although SHIFT button needed 

• Before calculators, logarithmic values had to be looked up in printed tables

What notation might I see with logarithms?

• 10 is a common base

  • log₁₀ x is abbreviated to log x or lg x

• The value e is another common base

  • log_e x is abbreviated to ln x

• (log x)² ≠ log x²

ln x

What is ln? 

• ln is a function that stands for natural logarithm

• It is a logarithm where the base is the constant "e"

  • $\ln x\equiv {\log }_{\mathrm{e}}x$

  • It is important to remember that ln is a function and not a number

• The natural logarithm ($\ln x$) and the exponential function (${\mathrm{e}}^x$) are inverses of each other 

• It is defined for all positive numbers ($x>0$)

  • $\ln x$ cannot be defined for negative numbers or $x=0$

What are the properties of ln? 

• Using the definition of a logarithm you can see

  • $\ln 1=0$

  • $\ln \mathrm{e}=1$

  • $\ln {\mathrm{e}}^x=x$ 

  • $\ln x$ is only defined for positive x

How can I solve equations involving e & ln? 

• The functions ${\mathrm{e}}^x$ and $\ln x$ are inverses of each other

  • If ${\mathrm{e}}^x=a$ then $x=\ln a$

  • If $\ln x=a$ then $x={\mathrm{e}}^a$

  • If ${\mathrm{e}}^{\mathrm{f}\left(x\right)}=\mathrm{g}\left(x\right)$ then $\mathrm{f}\left(x\right)=\ln \mathrm{g}\left(x\right)$

  • If $\ln \mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$ then $\mathrm{f}\left(x\right)={\mathrm{e}}^{\mathrm{g}\left(x\right)}$

• If your equation involves "e" then try to get all the "e" terms on one side

  • If "e" terms are multiplied, you can add the powers

    • ${\mathrm{e}}^x\times {\mathrm{e}}^y={\mathrm{e}}^{x+y}$ 

    • You can then apply ln to both sides of the equation

  • If "e" terms are added, try transforming the equation with a substitution

    • For example: If $y={\mathrm{e}}^x$ then ${\mathrm{e}}^{4x}=y^4$

    • You can then solve the resulting equation (usually a quadratic)

    • Once you solve for y then solve for x using the substitution formula

• If your equation involves "ln", try to combine all "ln" terms together

  • Use the laws of logarithms to combine terms into a single term

  • If you have $\ln \mathrm{f}\left(x\right)=\ln \mathrm{g}\left(x\right)$ then solve $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$

  • If you have $\ln \mathrm{f}\left(x\right)=k$ then solve $\mathrm{f}\left(x\right)={\mathrm{e}}^k$