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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e

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.

\color{blue}\ln(a)\color{grey}+\color{blue}\ln(b)\color{grey}=\color{blue}\ln(ab)

\color{blue}\ln(a)\color{grey}-\color{blue}\ln(b)\color{grey}=\color{blue}\ln\left(\dfrac{a}{b}\right)

\color{blue}\ln(a^{b})\color{grey}=\color{blue}b\ln(a)

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:

\color{red}e^{3x}\color{grey}=10

[2 marks]

\color{red}e^{3x}\color{grey}=10

3x=\color{blue}\ln(10)

\begin{aligned}x&=\dfrac{\color{blue}{\ln(10)}}{3}\\[1.2em]&=0.768\end{aligned}

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Example 2: Equations Involving Logarithms

Solve for x:

\color{blue}\ln(4x+3)\color{grey}=2

[2 marks]

\color{blue}\ln(4x+3)\color{grey}=2

4x+3=\color{red}{e^{2}}

4x=\color{red}e^{2}\color{grey}-3

\begin{aligned}x&=\dfrac{1}{4}(\color{red}e^{2}\color{grey}-3)\\[1.2em]&=1.10\end{aligned}

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The Exponential Function Example Questions

Question 1: Solve for x:

a) e^{x}=2

a) e^{5x}=19

c) e^{12x}=234

[6 marks]

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a) e^{x}=2

 

\begin{aligned}x&=\ln(2)\\[1.2em]&=0.693\end{aligned}

 

 

b) e^{5x}=19

 

5x=\ln(19)

 

\begin{aligned}x&=\dfrac{1}{5}\ln(19)\\[1.2em]&=0.589\end{aligned}

 

 

c) e^{12x}=234

 

12x=\ln(234)

 

\begin{aligned}x&=\dfrac{1}{12}\ln(234)\\[1.2em]&=0.455\end{aligned}

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Question 2: Solve for x:

a) \ln(x+1)=4

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

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

[6 marks]

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a) \ln(x+1)=4

 

x+1=e^{4}

 

\begin{aligned}x&=e^{4}-1\\[1.2em]&=53.6\end{aligned}

 

 

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

 

3x+2=e^{1.5}

 

3x=e^{1.5}-2

 

\begin{aligned}x&=\dfrac{1}{3}(e^{1.5}-2)\\[1.2em]&=0.827\end{aligned}

 

 

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

 

9x+36=e^{0.6}

 

9x=e^{0.6}-36

 

\begin{aligned}x&=\dfrac{1}{9}(e^{0.6}-36)\\[1.2em]&=-3.80\end{aligned}

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Question 3: Solve for x:

e^{2x}-13e^{x}+36=0

[4 marks]

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e^{2x}-13e^{x}+36=0

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

(e^{x})^{2}-13e^{x}+36=0

 

Substitute: y=e^{x}

y^{2}-13y+36=0

(y-9)(y-4)=0

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

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Question 4: Solve for x:

\ln(4x+3)-2\ln(x)=5

[5 marks]

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\ln(4x+3)-2\ln(x)=5

 

\ln(4x+3)-\ln(x^{2})=5

 

\ln\left(\dfrac{4x+3}{x^{2}}\right)=5

 

\dfrac{4x+3}{x^{2}}=e^{5}

 

4x+3=e^{5}x^{2}

 

e^{5}x^{2}-4x-3=0

 

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.

 

x=0.156
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Additional Resources

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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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