e & ln (Edexcel International A Level (IAL) Maths: Pure 3): Revision Note

"e"

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

"ln"

What is ln x? 

• 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

What properties of ln x do I need to know? 

• 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

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

  • $\ln a+\ln b=\ln \left(ab\right)$

  • $\ln a-\ln b=\ln \left(\frac{a}{b}\right)$

  • $n\ln a=\ln \left(a^n\right)$

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

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