Transcript C1: Differentiation from First Principles

C2: The Laws of Logarithms
Learning Objective: to be able to
use the laws of logarithms to rewrite expressions
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.
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)
The laws of logarithms
The division law
Let:
m = loga x
So:
x = am
and
n = loga y
and y = an
x am
= n

y a
Using the division law for indices:
x
= a mn
y
Writing this in log form gives:
x
m  n  loga
y
But m = loga x and n = loga y so:
x
loga x  loga y  loga
y
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
The Laws of Logarithms:
x
loga x  loga y  loga
y
n loga x = loga xn
loga x + loga y = loga (xy)
Also,
loga (1/x) = -loga x
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.
2loga 3 + loga 2  2loga 6 = loga 32 + loga 2  loga 62
= loga 9 + loga 2  loga 36
 9×2 
= loga 

36


= loga
1
2
The laws of logarithms
The laws of logarithms can also be used to break down a
single logarithm. For example:
2
Express log10
1
2
ab
in terms of log10 a, log10 b and log10 c.
4
c
2
1
2
1
ab
2 2
log10 4 = log10 a b  log10 c 4
c
1
2
2
= log10 a + log10 b  log10 c4
= 2log10 a + 21 log10 b  4log10 c
Logarithms to the base 10 are usually written as log or lg.
We can therefore write this expression as:
2log a + 21 log b  4log c
Task 1 :
• Exercise 3D