### m - Project Maths

```Logarithms
102 = 100
the base 10 raised to the power 2 gives 100
2 is the power which the base 10 must be raised to, to give 100
the power = logarithm
2 is the logarithm to the base 10 of 100
Logarithm is the number which we need to raise
a base to for a given answer
to what power must I raise 2 to give an answer of 64? ans = 6
written as log2 64 = 6
to what power must I raise 2 to give an answer of 64? ans = 6
written as log2 64 = 6
to what power must I raise 5 to give an answer of 625? ans = 4
written as log5 625 = 4
to what power must I raise 9 to give an answer of 3?
written as log9 3 = 1
2
loga n = p
ans = 1
2
ap = n
1. loga m + loga n = loga mn
2. loga m - loga n = loga m
n
3. n loga m = loga mn
4. logn m = loga m
loga n
change of base law
N.B. The log of a negative is impossible to find
Proofs: Law 1
loga m + loga n = loga m n
Let loga m = p & loga n = q
ap = m
aq = n
ap . aq = m . n
ap + q
= m.n
loga m . n = p + q
loga m n = loga m + loga n
Law 2
loga m - loga n = loga m
n
Let loga m = p & loga n = q
ap = m
ap
aq
aq = n
=
m
n
ap - q = m
n
loga m = p - q
n
loga m = loga m - loga n
n
Law 3
n loga m = loga mn
Let loga m = p
ap
= m
p )n
(a
apn
loga
mn
We need mn
=
( m )n
=
mn
= pn
loga mn = (loga m) n
loga mn = n loga m
Law 4
logn m = loga m
loga n
Let logn m = p
np = m
take logs of both sides
loga np = loga m
p loga n = loga m
p = loga m
loga n
logn m = loga m
loga n
e.g.1
log4 64 = x
x
4 = 64
4 x = 43
x = 3
e.g.2
log2 x = 5
25 = x
x = 32
e.g.3
log4 (5x + 6) = 2
= number inside
42 =
16 =
10 =
10 =
5
5x + 6
5x + 6
5x
x
2 = x
e.g.4
log3 (2x - 4) = 1 + log3 (4x + 8)
log3 (2x – 4) – log3 (4x + 8) = 1
 2x  4 
log3 
 1
 4x  8 
2x  4
3 
4x  8
1
3(4 x  8)  2 x  4
12x + 24 = 2x - 4
10x = -28
x = -28
10
x = -2.8
For an unknown power always
take logs of both sides
e.g.5
6n = 3200
log10 6n = log10 3200
n log10 6 = log10 3200
n = log10 3200
log10 6
n = 4.5045
Calculations using log10
log10 1000 = 3 as 103 = 1000
If we want log2 32 = log10 32
logn m = loga m
loga n
change of base law
log10 2
e.g.6
log2 55 = log10 x
log10 55 = log10 x
log10 2
5.78 = log10 x
105.78 = x
x = 604449