SOLUTION: what is the solution for 4 to the 2nd power x = 28

Question 408367: what is the solution for 4 to the 2nd power x = 28
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
What you posted was:

To solve we start by simplifying. Four squared is 16:
16x = 28
Divide by 16:

which reduces to:

But I suspect the equation is really

since you posted this under logarithms. If so this is
"4 to the 2x power = 28"
or
4^(2x) = 28

With the variable in an exponent we will need logarithms to solve for x. We can use a logarithm of any base. But if we use base 4 logarithms we will get the simplest answer (since it is a 4 being raised to the power). If we are looking for a decimal approximation then we would use logarithms our calculators "know": base 10 or base e (aka ln). I'll use base 4 logarithms:

On the left side we can use a property of logarithms, , to move the exponent of the argument out in front. (It is this very property that is the reason we use logarithms on problems like this. It allows us to move the exponent, where the variable is, out where we can now solve for the variable.) Using this property on your equation we get:

By definition . (This is why base 4 logarithms give us the simplest solution.) SO this simplifies to:

Now we just divide both sides by 2:

This is an exact expression for the solution to your equation. If you decide you want a decimal approximation for the answer, then you can either start over using base 10 or base e logarithms or you can just use the base conversion formula, , to convert the base 4 logarithm:

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