Logarithmic Functions (Cambridge International IGCSE Further Maths)

Logarithmic functions

What are logarithmic functions?

• A logarithm is the inverse of raising to a power

• If $a=b^{x}$ then $\mathrm{log}_{b}a=x$

  • $a>0$
  • $b$ is called the base of the logarithm

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

  • log _... end _ (... ) = would be read as “the power that you raise ... to, to get ..., is ”
  • So $log_{5} ( 125 ) = 3$ 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 $\mathrm{log}_{2}10$

How do I use logarithms?

• Recognising the rules of logarithms allows expressions to be simplified

• Recognition of common powers helps in simple cases For more awkward cases a calculator is needed

  • 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

• 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

  • log₁₀x is abbreviated to log x or lg x

• The value e is another common base

  • log_ex is abbreviated to ln x

• (log x)²≠ log x²

Exam Hint

• Before going into the exam, make sure you are completely familiar with your calculator and know how to use its logarithm functions

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"

  • $\mathrm{ln}x\equiv\mathrm{log}_{e}x$
  • It is important to remember that ln is a function and not a number

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

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

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

What are the properties of ln?

• Using the definition of a logarithm you can see

  • $\mathrm{ln}1=0$
  • $\mathrm{ln}e=1$
  • $\mathrm{ln}e^{x}=x$
  • $\mathrm{ln}x$ is only defined for positive x

How can I solve equations involving e & ln?

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

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

• 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

    • $e^{x}\times e^{y}=e^{x+y}$
    • 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 $y=e^{x}$ then $e^{4x}=y^{4}$
    • 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 $\ln \mathrm{f} ( x ) = \ln \mathrm{g} ( x )$ then solve $\mathrm{f} ( x ) = \mathrm{g} ( x )$
  • If you have $\ln \mathrm{f} ( x ) = k$ then solve $\mathrm{f} ( x ) = \mathrm{e}^{k}$

Exam Hint

• If you're working on the non-calculator exam paper you may need to leave answers as exact values so using ln*x *is a good idea

  • This can be solved with e^x which can then be left as an exact value

Worked Example Solve the equation $e^{2x}=5$, leaving your answer as an exact value.

Take the natural logarithm of both sides.

$\ln ( \mathrm{e}^{2 x} ) = \ln 5$

Use the property that $\ln ( \mathrm{e}^{a} ) = a$.

$2x=\mathrm{ln}5$* *

Divide both sides by 2.

$\begin{gathered} x & = \frac{\ln 5}{2} \end{gathered}$

Do not use your calculator to evaluate this as the question asks for the answer given as an exact value.

$x=\frac{\mathrm{ln}5}{2}$