Question 881537
you want to prove:


Log5(xy) = 2Log25(x) + 2Log25(y)


2Log25(x) is equal to Log25(x^2)
2Log25(y) is equal to Log25(y^2)


Log25(x^2) + Log25(y^2) is equal to Log25(x^2 * y^2) which is equal to Log25((xy)^2).


Your original equation becomes:


Log5(xy) = Log25((xy)^2)


set Log5(xy) equal to a.
set Log25((xy)^2) equal to b.


you get:


Log5(xy) = a
Log25((xy)^2) = b


If we can prove that a = b, then that proves that Log5(xy) = Log25((xy)^2) which proves the original equation is true.


The basic definition of logs states that:


Log5(xy) = a if and only if 5^a = xy


Likewise, the basic definition of logs states that:


Log25((xy)^2) = b if and only if 25^b = (xy)^2.


25^b is the same as (5^2)^b which makes this last equation equivalent to:


(5^2)^b = (xy)^2 which can also be expressed as:


5^(2b) = (xy)^2 which can also be expressed as:


(5^b)^2 = (xy)^2.


(5^b)^2 can only be equal to (xy)^2 if 5^b = xy.


you can prove this by just taking the square root of both sides of the equation and you will get 5^b = xy.


we now have:


5^a = xy
5^b = xy


This can only be true if a = b.


Since we have proven that a = b, we have also proven that Log5(xy) is equal to Log25((xy)^2) which proves the original equation is true.