Question 1118206
somehow these wind up being identical when x is positive.


the difference is that log4(x^2) allows the value of x to be positive or negative, while log2(x) allows the value of x to be positive only.


so, if you restrict your domain to positive values of x, the 2 equations are identical.


since they are identical, then x can be any positive value.


our equation is log2(x) = log4(x^2).


it's clear that x has to be positive, since otherwise log2(x) would not provide a real answer, since x has to be > 0.


log2(x) = a if and only if x^a = x.


that's from the basic definition of of what a log is.


likewise, log4(x^2) = b if and only if 4^b = x^2.


4 is equal to 2^2, therefore 4^b is equal to (2^2)^b and you get:


(2^2)^b = x^2


(2^2)^b is equal to 2^(2b) which is equal to (2^b)^2.


you get (2^b)^2 = x^2.


take the square root of both sides of the equaiton to get:


2^b = plus or minus x.


when 2^b = plus x, the basic definition of logs states that 2^b = x if and only if b = log2(x).


that's ok, since x is positive.


however, 2^b = -x leads to log2(-x) = b which can't be, since x has to be positive.


therefore, when x is posiive, you get:


2^a = x and you get 2^b = x.


this means that a must be equal to b.


the two equations become identical as long as x is positive.


so, you get:


log2(x) = a if and only if 2^a = x.


square both sides of that equation and you get (2^a)^2 = x^2


that's the same as 2^2a = x^2 which is the same as (2^2)^a = x^2 which is the same as 4^a = x^2.


the equations are identical as long as x is positive, therefore x can be any value as long as it is positive.


i'm not sure if i explained it well, but that's what i'm seeing that the answer is.


this can be seen in the following 3 graphs.


the first graph is y = log2(x).


the second graph is y = log4(x^2).


the third graph is both equations shown on the same graph.


you can see that the black line of y = log2(x) has turned red after it was superimposed  on by the orange line of y = log4(x^2).


that only happens on the right side of the graph because y = log2(x) is only valid when x is greater than 0.


y = log4(x^2) is valid for all real values of x except, i think, when x = 0.


neither graph is valid when x = 0.


here's the graphs.


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<img src = "http://theo.x10hosting.com/2018/060402.jpg" alt="$$$" >


<img src = "http://theo.x10hosting.com/2018/060403.jpg" alt="$$$" >