Exponential & Logarithmic Functions (DP IB Analysis & Approaches (AA): SL): Revision Note

An exponential function is defined by $f\left(x\right)=a^x,a>0$

Its domain is the set of all real values

Its range is the set of all positive real values

An important exponential function is $f\left(x\right)={\mathrm{e}}^x$

• Where e is Euler's constant 2.718…

Any exponential function can be written using e

• $a^x={\mathrm{e}}^{x\ln a}$

  • This is given in the formula booklet

The graphs have a y-intercept at $(0,1)$

The graph will always pass through the point $(1,a)$

The graphs do not have any roots

The graphs have a horizontal asymptote at the x-axis: $y=0$

• For $a>1$ this is the limiting value when x tends to negative infinity

• For $0<a<1$ this is the limiting value when x tends to positive infinity

The graphs do not have any minimum or maximum points

A logarithmic function is of the form $f\left(x\right)={\log }_{a}x,x>0$

Its domain is the set of all positive real values

• You can't take a log of zero or a negative number

Its range is set of all real values

${\log }_{a}x$ and $a^x$ are inverse functions

An important logarithmic function is $f\left(x\right)=\ln x$

• This is the natural logarithmic function $\ln x\equiv {\log }_{\mathrm{e}}x$

• This is the inverse of ${\mathrm{e}}^x$

  • $\ln {\mathrm{e}}^x=x$and ${\mathrm{e}}^{\ln x}=x$

Any logarithmic function can be written using ln

• ${\log }_{a}x=\frac{\ln x}{\ln a}$using the change of base formula

The graphs do not have a y-intercept

The graphs have one root at $(1,0)$

The graphs will always pass through the point $(a,1)$

The graphs have a vertical asymptote at the y-axis: $x=0$

The graphs do not have any minimum or maximum points