e & ln (Edexcel IAL Maths)

"e"

Spec point — 3.1

What is the number "e" in maths?

• e is an irrational number, sometimes called Euler's number

  • e ≈ 2.718

How do I sketch y = e^x?

• As other exponential graphs do, y = e^x

  • passes through (0, 1)
  • has the x-axis as an asymptote

What is the special property of "e"?

• y = e^x has the particular property

  • dy/dx = e^x
  • ie for every real number x, the gradient of y = e^x is also equal to e^x (see Derivatives of Exponential Functions)

How do I sketch y = e^-x?

• y = e^-x is a reflection in the y-axis of y = e^x

  • They are of the form y = f(x) and y = f(-x) (see Transformations of Functions - Reflections)

What is exponential growth and decay?

• y = Ae^kx (k > 0) is exponential growth
• y = Ae^-kx (k > 0) is exponential decay
• A is the initial value
• k is a (usually positive) constant
• “-“ is used in the equation making clear whether it is growth or decay

Worked Example

"ln"

Spec point — 3.2

What is ln x?

• 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

What properties of ln x do I need to know?

• 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

• As ln is a logarithm you can use the laws of logarithms

  • $\mathrm{ln}a+\mathrm{ln}b=\mathrm{ln}(ab)$
  • $\mathrm{ln}a-\mathrm{ln}b=\mathrm{ln}(\frac{a}{b})$
  • $n\mathrm{ln}a=\mathrm{ln}(a^{n})$

• Any logarithm can be written in terms of the natural logarithm using the change of base formula

  • $log_{a} b = \frac{\ln b}{\ln a}$

How do I solve equations involving e^x & ln x?

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

  • 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 $\mathrm{f} ( x ) = \mathrm{e}^{k}$

Worked Example

Exam Hint

• Always simplify your answer if you can

  • for example, $\frac{1}{2}\mathrm{ln}25=\mathrm{ln}\sqrt{25}=\mathrm{ln}5$
  • you wouldn't leave your final answer as $\sqrt{25}$ so don't leave your final answer as $\frac{1}{2}\mathrm{ln}25$