Transcript C1: Differentiation from First Principles

C2: Change of Base of
Logarithms
Learning Objective: to understand
that the base of a logarithm may
need changing to be able to solve
an equation
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
5x = 8
So:
Taking the log to the base 10 of both sides:
log 5x = log 8
So:
x log 5 = log 8
log 8
x=
log 5
log 8
log5 8 =
= 1.29 (to 3 s.f.)
log 5
Changing the base of a logarithm
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
logc b
x=
logc a
So:
logc b
loga b =
logc a
Example :
We can use the change of base of logarithms to solve
equations. For example:
Find, to 3 significant figures log8 11.
We can solve this by changing to base 10:
log8 11 = log10 11 / log10 8
Using a calculator:
x = 1.15 (to 3 s.f.)
Task 1 :
Find to 3 d.p.
• log7 120
• log3 45
• log2 19
• log11 3
• log6 4
Solve
8x = 14, 9x = 99,
12x = 6