SOLUTION: Given that x = 2ᵅ and y = 4ⁿ, show clearly that log₂(x²y) = 3α + 2n

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Question 1209209: Given that x = 2ᵅ and y = 4ⁿ, show clearly that log₂(x²y) = 3α + 2n
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Log Rules
log(M*N) = log(M)+log(N)
log(M^N) = N*log(M)
Let's refer to these as equations (1) and (2)

x+=+2%5Ea converts to the log form log%282%2C%28x%29%29+=+a
I'll refer to this as equation (3)

Also,
y+=+4%5En

y+=+%282%5E2%29%5En

y+=+2%5E%282n%29

log%282%2C%28y%29%29+=+2n Let's call this equation (4)

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Then we would write the following
log%282%2C%28x%5E2y%29%29+=+log%282%2C%28x%5E2%29%29%2Blog%282%2C%28y%29%29 Use equation (1)

log%282%2C%28x%5E2y%29%29+=+2%2Alog%282%2C%28x%29%29%2Blog%282%2C%28y%29%29 Use equation (2)

log%282%2C%28x%5E2y%29%29+=+2a%2Blog%282%2C%28y%29%29 Use equation (3)

log%282%2C%28x%5E2y%29%29+=+2a%2B2n Use equation (4)

I don't know where your teacher is getting the 3 in 3a.

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
Given that x = 2ᵅ and y = 4ⁿ, show clearly that log₂(x²y) = 3α + 2n
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This statement can not be proved,  since it is  highlight%28highlight%28FATALLY%29%29  highlight%28highlight%28WRONG%29%29. 

    To make sure that it is wrong, take a = 1, n = 1.

    Then  x%5E2 = %282%5E1%29%5E2 = 4;  y = 4%5E1 = 4;  x%5E2%2Ay = 4*4 = 16,

    log%282%2C%28x%5E2y%29%29 = log%282%2C16%29 = 4,


    but  3a + 2n = 3*1 + 2*1 = 3 + 2 = 5 --->  highlight%28highlight%28CONTRADICTION%29%29.

So,  this statement can not be proved - - - but it can be  highlight%28highlight%28DISPROVED%29%29,  instead.