Question 1003768
as far as i can tell, x = 1


when x = 1, the equation of:


(3x)^(1 + log3(x)) = 3 becomes:


(3*1)^(1 + log3(1)) = 3 which becomes:


(3)^(1 + log3(1)) = 3


log3(1) = y if and only if 3^y = 1
this can only occur if y = 0 because 3^0 = 1


therefore 1 + log3(1)) is equal to 1 + 0 which is equal to 1.


the equation becomes:


(3)^1 = 3 which becomes 3 = 3


the equation is true when x = 1.


how was this derived?


by magic of course.


here's the magic, if i did it correctly.


your equation is:


(3x)^(1+log3(x)) = 3


let y = 1 + log3(x) and your equation becomes:


(3x)^y = 3


this is true if and only if log3x(3) = y


log3x(3) is equal to log(3) / log(3x)


equation becomes log(3) / log(3x) = y


we had already set y = 1 + log3(x) 


log3(x) is equal to log(x)/log(3)


therefore:


y = 1 + log3(x) becomes:


y = 1 + log(x)/log(3)


now we have:


y = 1 + log(x) / log(3) and we have:


y = log(3) / log(3x)


y = 1 + log(x) / log(3) is equivalent to y = (log(3) + log(x)) / log(3)


y = log(3) / log(3x) is equivalent to y = log(3) / (log(3) + log(x))


replace y in the second equation with y = (log(3) + log(x)) / log(3) from the first equation to get:


(log(3) + log(x)) / log(3) = log(3) / (log(3) + log(x)).


multiply both sides of this equation by log(3) and multiply both sides of this equation by (log(3) + log(x)) to get:


(log(3) + log(x))^2 = log(3)^2


take the square root of both sides of this equation to get:


log(3) + log(x) = log(3)


subtract log(3) from both sides of this equation to get:


log(x) = 0


solve for x to get x = 1.


there may be another, easier way to solve this, but i couldn't figure it out.


this was quite convoluted and i'm happy i was able to get an answer that made sense the way that i did do it.


hopefully you can understand what i did and why i did it.


some concepts involved.


y = logb(x) if and only if b^y = x


b^y = x if and only if logb(x) = y


log(a*b) = log(a) + log(b)


log(a^b) = b*log(a)


logb(x) = loga(x) / loga(b)


a + b/c = (ac + b) / c