Transcript AS Maths Masterclass

AS Maths Masterclass
Lesson 5:
Exploring logarithms
Learning objectives
The student should be able to:
• relate indices to logarithms and solve simple
indicial equations;
• apply the laws of logarithms to simplify
mathematical expressions;
• solve simple logarithmic equations using the
laws of logarithms;
Why bother with logs ?
Thanks to John Napier/ Henry Briggs: 1615
… they were used to simplify very large
calculations in physics and astronomy.
But are logs actually used
nowadays ?
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Finance:
compound interest
Photography/ computing: image compression
Computer games: measuring algorithm efficiency
Cinema:
flicker in a motion picture
Music:
pitch, acoustics, sound intensity
Chemistry:
ph scale, Newton’s law of cooling
Physics:
radioactive decay, capacitor discharge
Biology/ Economics/ Geog.: population growth
Geology: earthquakes measured on the richter scale.
Recall indices from GCSE ?
Multiplication rule:
Division rule:
Power rule:
Reciprocal rule:
Power of zero:
Power of a product:
a a  a
m
n
m n
a a  a
(a )  a
m
m
n
mn
n
mn
1
a 
a
a 0  1 (a  0)
n
n
( a  b)  a  b
n
n
n
What is a logarithm?
The log of a number to a given base is the
index of the power to which the base must
be raised in order to obtain the original
number.
10  1000
3
e.g.
when
log 1000  3
10
If a  N then x is called
the log of N to the base a, written
x
x  log N
a
Common logs and Natural logs
If the base is 10 we say that the log is ..
common. We write “log” or just “lg”
If the base is e (=2.718 281 828 …) then we
say that the log is ..
natural. We write “ln”
Extension activity: investigate the number “e”.
Evaluating logarithms
x
x

log
N
If we write
instead of a  N
a
then we can easily find logs to any base
provided the number has a “nice”
relationship.
x
E.g. Let log 3 81  x then 3  81
Hence, x = 4 since 3.3.3.3 = 81
Click here to practice finding logs to any base (nice numbers)
Click here to view the Logarithm spreadsheet.
Moving towards the laws of logs
We all know that
so by definition,
a  1 (a  0)
0
log 1  0
a
In addition, we all know that a  a
and so again by definition, log a a  1
1
We now need to examine equivalent laws
for logs like we had for indices ……
The first law of logs
“The log of a product is the sum of the logs”
log xy  log a x  log a y
i.e. Prove that
a
Let u = log a x and v = log a y
Then, by definition we have x  a u and
u
v
u v
Hence,
xy  a .a  a
so by definition,
u + v = log a
Finally,
log x  log y  log xy
a
a
a
ya
xy
v
The power law
“The log of a number to a power is the product of
the power with the log of the number”
b
i.e. Prove that log a x  b. log a x
Let u = log a x so a  x and (a )  x
bu
b
Hence,
a x
b
log a x  b.u
so by definition,
u
u
From which,
b
log x  b. log x
b
a
a
b
Practice with the log laws
Let’s see how these laws work out in practice.
Teacher clicks here for demonstration of log laws with numbers
Students click here to practice the algebra of log laws
Extension activity: Click here for harder questions using the log laws
Changing the base
Whenever the numbers are not “nice”, we can always
change the base to one more convenient.
log x  log x
Prove
b
a
log a
b
Let u = log a x
Hence, log (a
so u log a  log
Finally,
log
u
b
b
and so a u  x
)  log x
log x
x meaning that u =
b
b
b
log a
log x
x  log a
a
b
b
b
Practice with change of base
Click here to practice change of base for "awkward logs"l
Click here for the IWB exercise on logs
Simple indicial equations
If, when solving equations we have a variable in the power,
then we say that we have an indicial equation: 22 x3  9.27
We can now solve indicial equations by either:
(a) changing the base; or
(b) taking logs and applying the power law:
Click here for some indicial equations using method (b)
Harder indicial and log equations
Three index terms is likely to lead to a
quadratic equation.
Click here to look at some quadratic indicial equations
Click here to solve logarithmic equations
Extension activity: Click here to try some harder log equations