The Exponential Function

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The Exponential Function Revision

The Exponential Function

We have met exponential functions before, but there is one specific exponential function that has special properties, and it is based around a special number: \color{red}e.

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The exponential function is \color{red}e^{x}.

\color{red}e=2.71828... is a number. It is a decimal that goes on forever
(like \pi).

\color{red}e^{x} has special properties, most notable being that the gradient of \color{red}e^{x} is \color{red}e^{x}. This will be very important in the differentiation section of the course.

There are some key facts to remember about the graph of y=e^{x}:

  • It crosses the y-axis at (0,1)
  • As x\rightarrow\infty, \color{red}e^{x}\color{grey}\rightarrow\infty and as x\rightarrow -\infty, \color{red}e^{x}\color{grey}\rightarrow0
  • \color{red}e^{x} is never negative.

y=e^{ax+b} + c is a transformation of y = e^x, where a is a horizontal stretch, b moves it horizontally and c moves it vertically.

y = e^{-x} reflects y=e^x in the y-axis.

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Natural Logarithm

The inverse function of \color{red}e^{x} is the natural logarithm \color{blue}\ln(x). This is the logarithm with base \color{red}e (\text{log}_e (x)).

All the laws of logarithms can be applied to the natural logarithm.




The graph of the natural logarithm (in blue) is the reflection in the line y=x of the graph of the exponential function (in red).

There are key facts to remember about the graph of \color{blue}y=\ln(x):

  • It crosses the x-axis at (1,0)
  • As x\rightarrow\infty, \color{blue}\ln(x)\color{grey}\rightarrow\infty and as x\rightarrow0, \color{blue}\ln(x)\color{grey}\rightarrow -\infty
  • \color{blue}\ln(x) does not take any values for x\leq0

Since \ln (x) is the inverse of e^x and is a logarithmic function, we have these formulas relating the two:

\textcolor{red}{e}^{\textcolor{blue}{\ln (x)}} = x

\textcolor{blue}{\ln} \textcolor{red}{(e^x)} = x

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Example 1: Equations Involving the Exponential Function

Solve for x:


[2 marks]




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Example 2: Equations Involving Logarithms

Solve for x:


[2 marks]





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The Exponential Function Example Questions

a) e^{x}=2





b) e^{5x}=19







c) e^{12x}=234





a) \ln(x+1)=4







b) \ln(3x+2)=1.5









c) \ln(9x+36)=0.6








Note that e^{2x}=(e^{x})^{2}



Substitute: y=e^{x}



y=9 or y=4


Reverse substitution:

e^{x}=9 or e^{x}=4

x=\ln(9) or x=\ln(4)

x=2.20 or x=1.39













Use quadratic formula:


\begin{aligned}x&=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\\[1.2em]&=\dfrac{4\pm\sqrt{(-4)^{2}-4\times e^{5}\times(-3)}}{2e^{5}}\\[1.2em]&=\dfrac{4\pm\sqrt{16+12e^{5}}}{2e^{5}}\\[1.2em]&=\dfrac{2\pm\sqrt{4+3e^{5}}}{e^{5}}\end{aligned}


x=0.156 or x=-0.129


We can discount the negative solution because \ln(x) is not valid for negative x.



Additional Resources


Exam Tips Cheat Sheet

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The Exponential Function Worksheet and Example Questions

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Exponentials and Natural Logarithms

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