Change of base formula

In logarithms, many times we encounter with the problem when there is a logarithm whose value is not known to us for that particular base, but we know its value for some other base.
For such cases we work by changing the base of logarithms by using Logarithmic Change of Base Formula which is easy to implement.

The Change of Base Formula for logarithm:
log a (x) = log b (x) / log b (a)
Here ‘a’ is base of logarithm of x before change and, b is base of logarithm of x after changing the base.
So above Log Change Base Formula can be used when you need to change base of logarithm from ‘a’ to b provided that the new base is useful to you, otherwise changing a base will not be useful.
Let us see proof of Logarithms Change of Base Formula and then we will go through some examples:

Proof:
We know that Raising ‘a’ (base of log a (x)) with the power of log a (x) gives x:
 x = a^(〖log〗_a  (x))………………………………………….(1)
Similarly, raising b with the power of log b (a) gives a:
a = b^(〖log〗_b  (a))        …………………………………………(2)
When we replace a in (1) with b^(〖log〗_b  (a)) given in equation (2), we get:
x = a^(〖log〗_(a ) (x)) =〖〖(b〗^(〖log〗_b  (a)))〗^(〖log〗_a  (x)) =  b^(〖log〗_b  (a)  〖* log〗_(a ) (x))     ………(3)
Applying log b() on both sides of equation (3) we get:
log b (x) = log b 〖(b〗^(〖log〗_b  (a)  〖* log〗_(a ) (x)))    
Using the log power rule we get:
log b (x) = log b (a)*log a(x)
Log a (x) = log b (x) / log b (a)
Now let us see some examples using the above formula:

Example 1) Change the base of following logarithm to 2: log12 64.
Solution) Using change of base formula this can be evaluated as:
log12 (64) = log2 (64)/log2 (12)

Example 2) Solve the following by changing base to e: log10ee
Solution) log10 ee = loge ee / loge 10
= e/ln 10 (loge 10 is written as ln 10)
= e/2.302   (as ln 10 =2.302; use calculator for this)

Example 3) Solve log9 27.
Solution) here it is difficult to find the value directly so we will change the base to 3 as 9 and 27 is square and cube numbers of 3 respectively.
Log 9 27 = log 3 27 / log3 9
= log3 33 / log3 32
= 3/2   (As loga bc = c(loga b) and also loga a = 1; these are other properties of logarithms)