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Solving Exponential Equations with the 'Same' Base - Problem 2
Solving an exponential equation when we have completely different bases. For this particular problem we're trying to solve for x, but yet we have a 16 to a power and an 8 to a power. If our bases are the same, we can just set our exponents equal, in this case it's not that easy.
What we need to think about is what base do 16 and 8 have in common. We can rewrite 16 as a power of 4, but we can't write 8 as a power of 4, so they both have to come down to 2. So rewriting both of these as a power of 2. I know that 16 is 2 to the fourth and I know that 8 is 2 to the third, so that's just rewriting what we have here, so our exponents are still hanging on outside.
So I haven't changed the problem at all I've just rewritten our bases as powers, so now we need simplify this up. Remember when you take a power to a power you multiply, so this is really the same thing as 2 to the 4 times x, or 2 to the 4x. Same thing on the other side power to power we multiply, so this becomes 2 to the, if we just say 3x minus 1 or if you want to rewrite that distribute it through 2 to the 3x minus 3, that's still the same thing.
So we have 2 to the 4x is equal to 2 to the 3x minus 3. Now all we do is look at our exponents, our bases are the same, so then our exponents have to be the same leaving us with 4x is equal to 3x minus 3, just solve for x subtract 3x, x is equal to -3.
So by rewriting both of our bases, we were able to get our bases the same and then just solve for x.