Exponential functions
Functions of the form 𝑓(𝑥)=&𝑎!, where 𝑎 is a constant, are called exponential functions. You should
become familiar with these functions and the shapes of their graphs.
For instance, table below shows an example of values for 𝑦 =&2!.
𝑥
-3
-2
-1
0
1
2
3
𝑦
1
8
1
4
1
2
1
2
4
8
The graph of 𝑦 =&2! is a smooth curve that looks like this:
&&&&&&&&
𝒚 = # 𝒆𝒙
Exponential functions of the form 𝑓(𝑥)=&𝑎! have a special pr operty. The graphs of their gradient
functions are a similar shape to the graphs of the function themselves. When the value of a is
approximately equal to 2.71878, the gradient function is exactly the same as the original function.
The exact value of this is represented by the letter e.
A similar result holds for functions such as 𝑒"!,𝑒#!and 𝑒!
"!.
Example 1: Differentiate with respect to 𝑥.
a. 𝑒$! b. 𝑒#!
"! c. 3𝑒%!
a. 𝑦 = 𝑒$!
𝑑𝑦
𝑑𝑥 =&4𝑒$!
b. 𝑦 = 𝑒#!
"!
𝑑𝑦
𝑑𝑥 =&−1
2𝑒#&
%!
c. 𝑦 = 3𝑒%!
𝑑𝑦
𝑑𝑥 = &2& ×3𝑒%! = 6𝑒%!
Exponential modelling
𝑒!&can be used to model situations such as population growth, where the rate of increase is
proportional to the size of the population at any given moment. Similarly, 𝑒#! can be used to model
radioactive decay, where the rate of decrease is proportional to the number of atoms remaining.
Exponentials and Logarithms Cheat Sheet
Example 2:
The density of a pesticide in a given section of field, P mg/m2 , can be modelled by the equation
𝑃 = 160𝑒#'.'')*
where t is the time in days since the pesticide was first applied.
a. Use this model to estimate the density of pesticide after 15 days.
After 15 days, 𝑡 = 15.
&&&&&&&&&&&&&&&&&&𝑃 = 160𝑒#'.'')×&"
&&&&&&&&&&&&&&&&&&𝑃 = 146.2&mg/m2
b. Interpret the meaning of the value 160 in this model.
When 𝑡 = 0,&&𝑃 = 160𝑒,=160, so 160 mg/m2 is the initial density of pesticide in the
field.
c. Show that -.
-* =𝑘𝑃,&where 𝑘 is a constant, and state the value of 𝑘.
𝑃 = 160𝑒#'.'')*
-.
-* = −0.96𝑒#'.'')*,&so 𝑘 = −0.96
d. Interpret the significance of the sign of your answer to part c.
As 𝑘 is negative, the density of the pesticide is decreasing (there is exponential decay)
e. Sketch the graph of P against t.
Logarithms
The inverses of exponential functions are called logarithms.
• log/&𝑛 = 𝑥 is equivalent to 𝑎!= 𝑛& (𝑎 ≠ 1)
Example 3: Write each statement as a logarithm.
a. 3!= 9 b. 2"=128 c. 64
!
"= 8
a. 3!= 9,+so log0&9 = 2
b. 2"=128, so log%&128 = 7
c. 64
!
"= 8, so log)$&8 = &
%
Laws of logarithms
Expressions involving more than one logarithm can be rearranged or simplified.
The laws of logarithms:
• log/&𝑥 +&log/&𝑦 =& log/&𝑥𝑦 (the multiplication law)
• log/&𝑥 −log/&𝑦 = log/&!
1 (the division law)
• log/&(𝑥2)= 𝑘log/&𝑥 (the power law)
You should also recognise the following special cases:
• log/
&
!=&log/&(𝑥#&)= −log/&𝑥 (the power law when&𝑘 = −1 )
• log/&𝑎 = 1 (𝑎 > 0,𝑎 ≠ 1)
• log/&1 = 0 (𝑎 > 0,𝑎 ≠ 1)
Example 4: Write as a singl e logarithm.
a. log0&6+&log0&7
=&log0&(6×7)
= &42
b. log%&15−&log%&3
=&log%&(15 ÷3)
=&log%&5
c.&&&2log"&3+ 3log"&2
2log"&3 =& log"&(3%)=log"&9
&&&&&&3log"&2 =&log"&(20)=log"&8
&&&&&&log"&9+&log"&8 = log"&72
d.&&&log&'&3−4log&' &F1
2G
&&&&&&4log&'&F1
2G = log&' &F1
2G$=log&' &F 1
16G
&&&&&&log&'&3−log&' &F 1
16G = log&' &F3 ÷ 1
16G =& log&'&48
Solving equations using logarithms
You can use logarithms and your calculator to solve equations of the form 𝑎!= 𝑏. You
can also solve more complicated equations by ‘taking logs’ of both sides.
• Whenever 𝑓(𝑥)= &𝑔(𝑥),& log/&𝑓(𝑥)=log/&𝑔(𝑥)&
Example 5: Solve the following equations, giving your answers to 3 decimal places.
a. 3!=20
So 𝑥 = &log0&20 = 2.727
b. 5$!#& =61
So 4𝑥 − 1 = &log"&61
&&&&&&&&&&&&&&4𝑥 = &log"&61+ 1
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&𝑥 = 34,5#3)&6&
$= 0.889
Working with natural logarithms
• The graph of 𝑦 = ln𝑥 is a reflection of the graph 𝑦 = 𝑒!&in the line 𝑦 = 𝑥.
The graph of 𝑦 = ln𝑥 passes through (1,0) and does not cross the y-axis.
The y-axis is an asymptote of the graph 𝑦 = ln𝑥. This means that ln𝑥 is only defined for
positive values of x.
Logarithms are the inverses of exponential functions. This rule can be used to solve
equations involving powers and logarithms.
• 𝑒78 !=ln(𝑒!)= 𝑥
• ln𝑥 = &log9𝑥
Example 6: Solve these equations, giving your answers in exact form.
a. 𝑒!= 5
When 𝑒!= 5
ln(𝑒!) = ln5
𝑥 = ln5
b. ln𝑥 = 3
When ln𝑥 = 3
𝑒78 != 𝑒0
𝑥 = 𝑒0
Logarithms and non-linear data
Logarithms can also be used to manage and explore non-linear trends in data.
If 𝑦 = 𝑎𝑥: then the graph of log𝑦 against log 𝑥 will be a straight line with gradient 𝑛 and
vertical intercept log𝑎.
Edexcel Pure Year 1
For all real values of 𝒙:
• If 𝒇(𝒙)=&𝒆𝒙&then 𝒇′(𝒙)=&𝒆𝒙
• If 𝒚 =&𝒆𝒙 then 𝒅𝒚
𝒅𝒙 =&𝒆𝒙
If
𝑦 = 𝑒2!
then
-1
-! =&𝑘𝑒2!
𝑦
𝑥
For all real values of 𝒙 and for any constant 𝒌:
• If 𝒇(𝒙)=&𝒆𝒌𝒙 &then 𝒇′(𝒙)=&𝒌𝒆𝒌𝒙
• If 𝒚 =&𝒆𝒌𝒙 then 𝒅𝒚
𝒅𝒙 =&𝒌𝒆𝒌𝒙
Use the rule for differentiati ng 𝑒2! with 𝑘 = 4
To differentiate 𝑎𝑒2! , multiply the whole
function by 𝑘.&The derivate is 𝑘𝑎𝑒2!.
𝑃
𝑡
0
160
Logarithms can take fractional or negative values
Use the log button on your calculator
You can write the natural logarithm on both sides
log&𝑦
log𝑥
0
log𝑎
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