Laws of Logarithms (OCR AS Maths A: Pure): Revision Note

Laws of logarithms

What are the laws of logarithms?

• There are many laws or rules of indices, for example

  • a^m x aⁿ = a^m+n

  • (a^m)ⁿ = a^mn

• There are equivalent laws of logarithms (for a > 0)

  • ${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$

  • ${\log }_a\left(\frac{x}{y}\right)={\log }_{a}x-{\log }_{a}y$

  • ${\log }_a{x}^k=k{\log }_{a}x$ 

• There are also some particular results these lead to

  • ${\log }_{a}a=1$

  • ${\log }_a{a}^x=x$

  • $a^{{\log }_{a}x}=x$

  • ${\log }_{a}1=0$

  • ${\log }_a\left(\frac{1}{x}\right)=-{\log }_{a}x$

• Two of these were seen in the notes Logarithmic Functions

• Beware …

  • … log (x + y) ≠ log x + log y

• Results apply to ln too

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

  • In particular ${\mathrm{e}}^{\ln x}=x$ and $\ln \left({\mathrm{e}}^x\right)=x$

How do I use the laws of logarithms?

• Laws of logarithms can be used to …

  • … simplify expressions

  • … solve logarithmic equations

  • … solve exponential equations

   

  

"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 are properties of ln x? 

• 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

  • $\ln a+\ln b=\ln \left(ab\right)$

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

  • $n\ln a=\ln \left(a^n\right)$

How are e^x and ln x inverses of each other? 

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