The Base Change Rule in Logarithms

5. Important rule on change of base in logarithms:

Remember the following important rule related to change of base in logarithms

for example:
log 6 7 = (log 5 7)/(log 5 6)

6. Rule of Base Change

How to change base of logarithms:

log a n = log b n . log a b

Without going into the proof and derivation of the rule, let us see this rule in a simple way taking the help of the formula:

log a b = (log b)/(log a)

log b n . log a b

= [(log n)/(log b)].[(log b)/(log a)]

= log a n

So, log a n = log b n . log a b

Let us now solve a few problems based on the rule no. 3 and rule no. 4 on Base Change

1. Prove the following

1. log 64 3 = (log 4 3)/3
2. log pq a = (log p a)/(1 + log p q)


We will convert all logs into base 4, as we see log to base 4 in the right side in question no.

1. So,
log 64 3 = (log 4 3)/(log 4 64)

from Law 3 on laws of logarithms, we have

log a (p)n = n log a p

so, log 4 4 3 = 3log 44

= 3 . 1 = 3, since log 44 = 1

Therefore, (log 4 3)/ (log 4 64)

= (log 4 3)/3

Solution 2:

we will convert all the logs into base p as we see base p on the right side.

log pq a = (log p a)/(log p pq)

Now, log p pq = log p p + log p q

= 1 + log p q


log pq a = (log p a)/ (1 + log p q)

7. Common Logarithms:

Logarithms expressed or calculated to base 10 are called Common Logarithms

For eg, log 10 5, log 10 p are examples of

common logarithms.

Common logarithms such as log 10 x, log 10a

can also be merely expressed as log x or log a implying that the base 10 is understood and therefore need not be shown.

This holds only for logarithms to base 10.

For logarithms to any base other than 10, the base must be clearly mentioned. For example, if the base is 5, then it should be clearly mentioned as log5 x.

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