SOLUTION: 2 raised to x+1 plus 2 raised to x-1 = 1280, what is the value of x?

Question 214629: 2 raised to x+1 plus 2 raised to x-1 = 1280, what is the value of x?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

The key to this problem is to get the variable into a single exponent. In order to do this we need to find the GCF of the two terms on the left. Both terms are made up of factors of 2. The first term has (x+1) 2's as factors and the second term has (x-1) 2's as factors. Since (x-1) is smaller (by two, to be exact) than (x+1), the GCF is (x-1) 2's. Factoring this GCF out we get:

(If this factoring is not clear to you, multiply it back out using the distributive property and see how you end up back where you started.)
Now that we have the variable in just one exponent, the equation is now relatively easy to solve. Start by simplifying:

Divide both sides by 5:

Now, in general, we use logarithms to solve equations like this. But, if we recognize that , we can solve it more easily:

If this is true then
x - 1 = 8
Adding 1 to both sides we get:
x = 9

Here's the logarithm solution to

(You could use natural logarithms (ln) instead.)
A property of logarithms is . Using this on our equation we get:

Dividing both sides by log(2):

Adding 1 to both sides we get:

Now, if we use calculators to find the logarithms, we should be able to simplify this. And, believe it or not, we get and answer of 9!
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