The Exponential Function

The Exponential Function

A LevelAQAEdexcelOCRAQA 2022Edexcel 2022OCR 2022

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.

A Level AQA Edexcel OCR

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.

A LevelAQAEdexcelOCR

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

A LevelAQAEdexcelOCR
A Level AQA Edexcel OCR

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}

A LevelAQAEdexcelOCR

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}

A LevelAQAEdexcelOCR

Example Questions

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}

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}

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

\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

Additional Resources

MME

Exam Tips Cheat Sheet

A Level
MME
A Level

Worksheet and Example Questions

Site Logo

Exponentials and Natural Logarithms

A Level

You May Also Like...

A Level Maths Revision Cards

The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams.

£14.99
View Product

A Level Maths – Cards & Paper Bundle

A level maths revision cards and exam papers for Edexcel. Includes 2022 predicted papers based on the advance information released in February 2022! MME is here to help you study from home with our revision cards and practise papers.

From: £22.99
View Product

Transition Maths Cards

The transition maths cards are a perfect way to cover the higher level topics from GCSE whilst being introduced to new A level maths topics to help you prepare for year 12. Your ideal guide to getting started with A level maths!

£8.99
View Product