Understanding Logarithmic Properties: Explained with Examples

[Rafe]

Okay i did a search for logarithmic properties and logarithms and couldn't seem to find an explanation for how this particular property works.
(log base c of a ) ^ b = b (log base c of a)
when i input simple numbers like :

a=4
b=3
c=2

Log base 2 of 4 obvioussly the answer is 2, but
2^3 /= (does not equel) 3 x 2.
i dont' know how to make sense of this discrepency. i imagine I'm just reading it wrong.

 

[fargoth]

hmmm the right formula is $$log_c(a)^b=blog_c(a)$$

edit: heh, I am tired =P

 

[fargoth]

because $$(c^a)^b=c^{a*b}$$.

 

[StatusX]

Actually the correct formula is:

$$\log_c(a^b) = b\log_c (a)$$

This can be proven by taking the base c exponential of each side:

$$c^{\log_c(a^b)} = a^b$$

$$c^{b\log_c (a)} = (c^{\log_c (a)})^b= (a)^b$$