47: more logarithms and indices © christine crisp “teach a level maths” vol. 1: as core modules
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47: More Logarithms 47: More Logarithms and Indicesand Indices
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
More Logarithms and Indices
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Module C2
More Logarithms and Indices
We need to be able to change between index forms for numbers and log forms.
bxba ax log
We use
We’ll also develop some more laws of logs.
More Logarithms and Indices
6443
e.g. Write the following in a form using logarithms:
279 23
(a)
(b)
6443 (a) 64log3 4
27log923
e.g. Write the following without using logarithms: 481log 3
21
4 2log (a)
(b)
Solution: (a)
481log 3 8134
24 21
279 23
(b)
21
4 2log (b)
The index, 3, is the log of 64 and 4 is the base.
Solution: bxba ax log
More Logarithms and Indices
932
1. Write the following in a form using logarithms:
416 21
(a)
(b)
Solution: 1(a)
932 9log2 3
4log1621
2. Write the following without using logarithms:
3125log5 23
4 8log (a)
(b)
12553
84 23
3125log5 2(a)
Exercises
416 21
(b)
23
4 8log (b)
More Logarithms and Indices
Some logs can be simplified.
Simplifying Logs
9log 3e.g. 1 Simplify
239
This log can be simplified because we can write 9 in index form using the base 3.
So, 9log 32
3 3log
The base, 3, is now the same as the base of the log
2 since a log is an index!
We are not solving an equation!
ka ka logIn
general,
More Logarithms and Indices
Simplifying Logs
42 2log
16log 2e.g. 2. Simplify (a) (b)
3
1log9
21
9
1log9
21
9log9
21
(b)
3
1log9
16log 2Solution: (a)
4
9
1log9
More Logarithms and Indices
100log10
1. Simplify the following log expressions:(a
)(b)
210 10log
5log5 81
2log(c)
(d)
64log4
Solution (a) 100log10
2
(b)
64log4
)4(log 34
3
(c)
5log521
5log5
21
81
2log(d)
322
1log
32 2log 3
Exercises
More Logarithms and Indices
There are 2 special cases we can get directly from the definition of a log.
Let x = 0,
2 useful results
bxba ax log
ba 0 1b
So, 1log010aa
By the law of indices,
for all values of the base01log a
More Logarithms and Indices
There are 2 special cases we can get directly from the definition of a log.
Let b = a,
x = 1
ax alogaa x Then
2 useful results
bxba ax log
More Logarithms and Indices
aa x
There are 2 special cases we can get directly from the definition of a log.
Let b = a,
x = 1
1Then
aalog
So, 1log aa
2 useful results
bxba ax log
More Logarithms and Indices
SUMMARY
bxba ax log
1log aa
Three Laws of Logarithms
The Definition of a Logarithm
01log a
ka ka log
More Logarithms and Indices
1. Simplify the following: 4log4
1910log10b
a alog
(a)
(c)
(b) 1log 2
(d)
(a) 1Ans:
(b) 0
(c) 19 (d) b
Exercises
More Logarithms and Indices
More Logarithms and Indices
The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.
More Logarithms and Indices
6443
e.g. Write the following in a form using logarithms:
279 23
(a)
(b)
6443 (a) 64log3 4
27log923
e.g. Write the following without using logarithms: 481log 3
21
4 2log (a)
(b)
Solution: (a)
481log 3 8134
24 21
279 23
(b)
21
4 2log (b)
Solution: bxba ax log
More Logarithms and Indices
Simplifying Logs
42 2log
16log 2e.g. 2. Simplify (a) (b)
3
1log9
21
9
1log9
21
9log9
21
(b)
3
1log9
16log 2Solution: (a)
4
9
1log9
More Logarithms and Indices
There are 2 special cases we can get directly from the definition of a log.
Let x = 0,
2 useful results
bxba ax log
ba 0 1b
So, 1log010aa
By the law of indices,
for all values of the base01log a
More Logarithms and Indices
aa x Let b = a,
x = 1
1Then
aalog
So, 1log aa
bxba ax log
More Logarithms and Indices
bxba ax log
1log aa
Three Laws of Logarithms
The Definition of a Logarithm
01log a
ka ka log
SUMMARY