Question 418638: 
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
Solving equations where the variable is in an exponent is often done with logarithms. And this equation can be solvde that way, too.
However, if it is possible to rewrite each side of the equation as powers of the same number, then there is an easier way than using logarithms. Since 9, 27 and 81 are all powers of 3, we will be able to rewrite each side as a power of 3:
Next, we will replace the square root with an exponent of 1/2:
There are two instances of powers of a power. The rule for the exponents when raising a power to a power is to multiply the exponents. So:
which simplifies to:
Next to multiply the powers of 3 on the left side, the rule for the exponents is to add the exponents. So:
which simplifies as follows:
We now have both sides of the equation written as powers of 3. The only way these powers of 3 can be equal is for the exponents to be equal, too. So:
This is a simple equation to solve. Just divide by 20 (or multiply by 1/20):
This is the solution to your equation.
Note: Using logarithms will work. But it will be much harder to find that 7/40 is the solution. So when you solve equations like this, where the variable is in an exponent, look to see if each side of the equation is a product, quotiemt or power of numbers that are powers of the same number. The method above is almost always easier to use. Use logarithms when you can't write each side as powers of the same number.
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