document.write( "Question 410264: (1/4)^(2x)=(1/2)^(x) \n" ); document.write( "

Algebra.Com's Answer #288569 by jsmallt9(3758) $\"\"$ $\"About$

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$\"%281%2F4%29%5E%282x%29+=+%281%2F2%29%5E%28x%29\"$
\n" ); document.write( "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.

\n" ); document.write( "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:
\n" ); document.write( " $\"%28%281%2F2%29%5E2%29%5E%282x%29+=+%281%2F2%29%5E%28x%29\"$
\n" ); document.write( "On the left side the rule for exponents when raising a power to a power is to multiply the exponents:
\n" ); document.write( " $\"%281%2F2%29%5E%282%2A2x%29+=+%281%2F2%29%5E%28x%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"%281%2F2%29%5E%284x%29+=+%281%2F2%29%5E%28x%29\"$
\n" ); document.write( "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:
\n" ); document.write( "4x = 2x
\n" ); document.write( "Subtracting 2x from each side we get:
\n" ); document.write( "2x = 0
\n" ); document.write( "Dividing both sides by 2 we get:
\n" ); document.write( "x = 0
\n" ); document.write( "This is the solution to your equation. \n" ); document.write( "

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