1. AS-Level Maths: Core 2 for Edexcel
C2.6 Exponentials and logarithms
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2. Exponential functions
Logarithms
The laws of logarithms
Solving equations using logarithms
Examination-style questions
Contents
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3. Exponential functions
So far in this course we have looked at many functions involving terms in xn.
In an exponential function, however, the variable is in the index. For example:
y = 2x
y = 5x
y = 0.1x
y = 3–x
y = 7x+1
The general form of an exponential function to the base a is:
y = ax where a > 0 and a ≠1.
Explain that exponential functions frequently occur naturally. For example, when cells multiply by doubling or when radioactive materials decay.
Discuss why negative values of a , a = 0 and a = 1 are not included in the general definition of an exponential function. For example, if we tried to plot y = (–2)x there would be values of x, for example x = ½, for which the function could not be evaluated. Also if a were equal to 1 the function would be a straight line (y = 1).
You have probably heard of exponential increase and decrease or exponential growth and decay.
A quantity that changes exponentially either increases or decreases increasingly rapidly as time goes on.

4. Graphs of exponential functions
Modify the graph and observe what happens to the curve.
Note that all of the graph will always pass through the point (0, 1) and ask students to give you the reason for this.
Establish that when a is greater than 1 the graph shows an exponential increase for positive values of x. When a is between 0 and 1, the graph shows an exponential decrease for positive values of x. Ask students to explain why the graph will never touch the x-axis.

5. Exponential functions
When a > 1 the graph of y = ax has the following shape:
When 0 < a < 1 the graph of
y = ax has the following shape: y x y 1 1
(1, a)
(1, a) x
In both cases the graph passes through (0, 1) and (1, a).
Tell students that the graph on the left shows exponential growth while the graph on the right shows exponential decay.
This is because:
a0 = and a1 = a
for all a > 0.

6. Contents Logarithms Exponential functions Logarithms
The laws of logarithms
Solving equations using logarithms
Examination-style questions
Contents
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7. Logarithms
Find p if p3 = 343
We can solve this equation by finding the cube root of 343:
Now, consider the following equation:
Find q if 3q = 343
We need to find the power of 3 that gives 343.
Ask students to use the xy key on their calculators to try to find q if 3q = 343.
Using trial and improvement they should find that q lies between 5.31 and By trying they should conclude that q = 5.31 to 2 decimal places.
One way to tackle this is by trial and improvement.
Use the xy key on your calculator to find q to 2 decimal places.

8. Logarithms
To avoid using trial and improvement we need to define the power y to which a given base a must be raised to equal a given number x.
This is defined as:
y = loga x
“y is equal to the logarithm, to the base a, of x”
The expressions
y = loga x
ay = x
and
are interchangeable.
This can be written using the implication sign :
Explain that the exact value can be found using logarithms and define logax as the power to which a must be raised to give x.
The method for finding the value of log3 343 using a calculator is explained later in this presentation.
y = loga x  ay = x
For example, 25 = 32 can be written in logarithmic form as:
log2 32 = 5

9. Logarithms Taking a log and raising to a power are inverse operations.
We have that:
y = loga x  ay = x
So:
Also:
y = loga ay
In the first example we are raising 7 to the power that 7 must be raised by to give 2. This must be 2.
The second example is the power to which 3 must be raised to give 36, so this must be 6.
For example: 2
and 6

10. Contents The laws of logarithms Exponential functions Logarithms
Solving equations using logarithms
Examination-style questions
Contents
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11. Some important results
When studying indices we found the following important results:
a1 = a
This can be written in logarithmic form as:
loga a = 1
a0 = 1
This can be written in logarithmic form as:
loga 1 = 0
It is important to remember these results when manipulating logarithms.

12. The laws of logarithms
The laws of logarithms follow from the laws of indices:
The multiplication law
Let:
m = loga x and n = loga y
So:
x = am and y = an 
xy = am × an
Using the multiplication law for indices:
xy = am + n
Writing this in log form gives:
m + n = loga xy
But m = loga x and n = loga y so:
loga x + loga y = loga (xy)

13. The laws of logarithms The division law Let: m = loga x and n = loga y
x = am and y = an 
Using the division law for indices:
Writing this in log form gives:
But m = loga x and n = loga y so:

14. The laws of logarithms The power law Let: m = loga x So: x = am 
xn =(am)n
Using the power law for indices:
xn =amn
Writing this in log form gives:
mn = loga xn
But m = loga x so:
n loga x = loga xn

15. The laws of logarithms
These three laws can be used to combine several logarithms written to the same base. For example:
Express 2loga 3 + loga 2 – 2loga 6 as a single logarithm.

16. The laws of logarithms
The laws of logarithms can also be used to break down a single logarithm. For example:
Express log in terms of log10 a, log10 b and log10 c.
Logarithms to the base 10 are usually written as log or lg.
We can therefore write this expression as:

17. Logarithms to the base 10 and to the base e
Although the base of a logarithm can be any positive number, there are only two bases that are commonly used.
These are:
Logarithms to the base 10
Logarithms to the base e
Logarithms to the base 10 are useful because our number system is based on powers of 10.
They can be found by using the log key on a calculator.
Tell students that the number e is irrational and its value is approximately The functions y = ex and y = ln x will be discussed later in the course.
Logarithms to the base e are called napierian or natural logarithms and have many applications in maths and science.
They can be found by using the ln key on a calculator.

18. Changing the base of a logarithm
Suppose we wish to calculate the value of log5 8.
We can’t calculate this directly using a calculator because it only find logs to the base 10 or the base e.
We can change the base of the logarithm as follows:
Let x = log5 8
So:
5x = 8
Taking the log to the base 10 of both sides:
log 5x = log 8
Ask students to find the value of log 8 ÷ log 5 using a calculator. This is the power to which 5 must be raised to give 8.
x log 5 = log 8
So:
1.29 (to 3 sig. figs.)

19. Changing the base of a logarithm
If we had used log to the base e instead we would have had:
1.29 (to 3 sig. figs.)
In general, to find loga b:
Let x = loga b, so we can write
ax = b
Taking the log to the base c of both sides gives:
logc ax = logc b
xlogc a = logc b
Ask students to find the value of ln 8 ÷ ln 5 using a calculator and to verify that this gives the same answer as log 8 ÷ log 5.
So:

20. Solving equations using logarithms
Exponential functions
Logarithms
The laws of logarithms
Solving equations using logarithms
Examination-style questions
Contents
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21. Solving equations involving logarithms
We can use the laws of logarithms to solve equations. For example:
Solve log5 x + 2 = log5 10
To solve this equation we have to write the constant value 2 in logarithmic form:
2 = 2 log because log5 5 = 1
= log5 52
= log5 25
The equation can now be written as:
log5 x + log5 25 = log5 10
log5 25x = log5 10
25x = 10
x = 0.4

22. Solving equations of the form ax = b
We can use logarithms to solve equations of the form ax = b. For example:
Find x to 3 significant figures if 52x = 30.
We can solve this by taking logs of both sides:
log 52x = log 30
2x log 5 = log 30
Tell students to avoid using a calculator until the very last stage. Since the numbers involved are not exact, this avoids errors due to rounding.
This equation could also be solved using logs to the base e. Students could check that this gives the same solution. The important thing to remember is to use the same base consistently throughout the equation.
Using a calculator:
x = 1.06 (to 3 sig. figs.)

23. Solving equations of the form ax = b
Find x to 3 significant figures if 43x+1 = 7x+2.
Taking logs of both sides:

24. Solving equations of the form ax = b
Solve 32x –5(3x) + 4 = 0 to 3 significant figures.
If we let y = 3x we can write the equation as:
So:
If 3x = 1 then x = 0.
Now, solving 3x = 4 by taking logs of both sides:
Tell students that this is a quadratic equation in 3x.

25. Examination-style questions
Exponential functions
Logarithms
The laws of logarithms
Solving equations using logarithms
Examination-style questions
Contents
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26. Examination-style question
Julia starts a new job on a salary of £ per annum. She is promised that her salary will increase by 4.5% at the end of each year. If she stays in the same job how long will it be before she earns more than double her starting salary?
× 1.045n =
1.045n = 2
log 1.045n = log 2
n log = log 2
15.7
Julia’s starting salary will have doubled after 16 years.