log a ( x ^ (b) ) = b log a (x)
we need to relate x to the base a. So, we let x = a ^ (n). Then,
log a ( x ^ (b) ) = log a ( ( (a) ^ n ) ^b )
= log a ( (a) ^ (n * b) )
Assumption that x = a ^ (n)
n = log a (x)
Therefore
log a (x ^ (b) ) = log a ( a ^ ( b * log a (x) ) )
= b * log a (x)
For example, we can show that log 2 ( (3) ^ 5 ) = 5 log 2 (3).
log 2 ( (3) ^ 5 ) = log 2 (3*3*3*3*3)
= log 2 (3) + log 2 (3) + log 2 (3) + log 2 (3) + log 2 (3)
= 5 log 2 (3)
Therefore, log 2 ( (3) ^ 5 ) = 5 log 2 (3)
For example, we can show that log 2 ( (5) ^ 2 ) = 2 log 2 (5).
log 2 ( (5) ^ 2 ) = log 2 (5*5)
= log 2 (5) + log 2 (5)
= 2 log 2 (5)
Therefore, log 2 ( (5) ^ 2 ) = 2 log 2 (5)
