Question 723738: 4^7x-1=16
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! First of all, please put exponents (especially multiple-term ones) in parentheses. What you posted meant
but what I suspect you intended was:
If I am wrong and you really meant what you posted, you will have to re-post your question because I'm going to solve what I think the problem is. If you re-post then I suggest that you post it as:
(4^7x)-1=16
with the parentheses making it completely clear that the -1 is not in the exponent.
Any equation where the variable is in the exponent can be solved using logarithms. But if the equation can be written so that each side is a power of the same number, then there is a much faster, easier, more exact way to solve the equation. It is definitely worth the effort to see if this alternate method can be used.
First we need a single term on each side. We have that already. Now we ask ourselves: "Can 4 be written as a power of 16? Or can 16 be written as a power of 4? Or can both 4 and 16 be written as a power of some other number?" If we can say "yes" to any of these questions, then we may use the faster method. With some thought we should be able to answer "yes" to all three! (We only needed one "yes".)  (think about what an exponent of 1/2 means and this will make sense),  ,  and  .
The faster method is:- Rewrite the equation so that both sides are powers of the same number.
- The only way that two powers of the same number can be equal (as the equation now says they are), is if the exponents are equal, too. So write a new equation that says that one exponent equals the other.
- Solve this new equation.
Let's see this in action:
1. Rewrite the equation.
I'm going to replace the 16 with  so that each side is a power of 4:
2. Write the equation with the exponents.
7x - 1 = 2
3. Solve
Adding 1:
7x = 3
Divide by 7:
x = 3/7
|
|
|
