Question 624424: If 8^1+x = 16^x , then x equals
a) 1/2 b)2/7 c)1/4 d)2/3 e) 2/5 f)3
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! What you posted means:
but I suspect it is supposed to be:
If I am right, then please use parentheses around exponents, especially multiple-term ones. If I am wrong, you will have to re-post your question.
When an equation has the variable in the exponent, logarithms are often used to solve the equation. And logarithms can certainly be used here.
However if you are able to write the equation in the form  , which says that two powers of the same number are equal, then there is a much easier way. While 8 is not a (well-known) power of 16 and 16 is not a (well-known) power of 8, they are both powers of 2! So we can solve your equation this faster, easier way.
Since  and  we can rewrite your equation as:
Now we can use the power of a power rule for exponents (i.e. multiply the exponents):
which simplifies to:
Now that we have the desired form, the next step is based on some simple logic: The only way two powers of 2 can be equal is if the exponents are equal. IOW, there is no way for two different powers of 2 to result in the same number. From this logic we know that the exponents are equal:
3 + 3x = 4x
This is very easy to solve. Just subtract 3x from each side:
3 = x
If you use logarithms you end up with the same answer. Although you may, because of decimal approximations, get an answer like 2.99999996 or 3.0000003 instead the exact answer of 3. So not only is the method we used faster and easier, it is also avoids small errors due to calculators approximating logarithms.
Some equations, like  , have to be solved with logarithms because the two bases are not (well-known) powers of each other nor are they (well-known) powers of some third number (like 8 and 16 were).
|
|
|
