SOLUTION: (1/4)^(2x)=(1/2)^(x)

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Question 410264: (1/4)^(2x)=(1/2)^(x)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%281%2F4%29%5E%282x%29+=+%281%2F2%29%5E%28x%29
With equations where the variable is in an exponent, you will usually solve them in either of two ways:
  • Rewrite both sides as powers of the same base. This is the easier path but it is not always possible to write the equation this way.
  • Use logarithms.
The easier way is so much easier it is worth taking time to see if you can find a way to rewrite both sides of the equations as powers of the same number.

Your equation can be done the easy way. You just have to notice that 1%2F4+=+%281%2F2%29%5E2. So we can rewrite each side as a power of 1/2:
%28%281%2F2%29%5E2%29%5E%282x%29+=+%281%2F2%29%5E%28x%29
On the left side the rule for exponents when raising a power to a power is to multiply the exponents:
%281%2F2%29%5E%282%2A2x%29+=+%281%2F2%29%5E%28x%29
or
%281%2F2%29%5E%284x%29+=+%281%2F2%29%5E%28x%29
We now have each side of the equation as powers of 1/2. The only way for these powers of 1/2 to be equal is if the exponents themselves are equal. So:
4x = 2x
Subtracting 2x from each side we get:
2x = 0
Dividing both sides by 2 we get:
x = 0
This is the solution to your equation.