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The Exponential Function
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.
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
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}
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}
The Exponential Function Example Questions
Question 1: Solve for x:
a) e^{x}=2
a) e^{5x}=19
c) e^{12x}=234
[6 marks]
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}
Question 2: Solve for x:
a) \ln(x+1)=4
a) \ln(3x+2)=1.5
a) \ln(9x+36)=0.6
[6 marks]
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}
Question 3: Solve for x:
e^{2x}-13e^{x}+36=0[4 marks]
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
Question 4: Solve for x:
\ln(4x+3)-2\ln(x)=5[5 marks]
\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
Specification Points Covered
F1 – Know and use the function a^x and its graph, where a is positive
Know and use the function e^x and its graph
F2 – Know that the gradient of e^{kx} is equal to ke^{kx} and hence understand why the exponential model is suitable in many applications
The Exponential Function Worksheet and Example Questions
Exponentials and Natural Logarithms
A Level