SOLUTION: 2^X=1/8

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Question 280972: 2^X=1/8
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2%5Ex+=+1%2F8
The quick solution to this one is to answer the question: "What power of 2 is 1/8?". If you understand exponents and powers of 2 this is a question you can answer.

Otherwise we can solve this using logarithms. To get the simplest exact expression for the solution we will use base 2 logarithms:
log%282%2C+%282%5Ex%29%29+=+log%282%2C+%281%2F8%29%29
On the left side we can use the property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent out in front. This property and its ability to move the exponent out in front is the very reason we use logarithms on problems like this: to get the variable out of the exponent.
x%2Alog%282%2C+%282%29%29+=+log%282%2C+%281%2F8%29%29
Since log%282%2C+%282%29%29+=+1 be definition the left side simplifies to:
x+=+log%282%2C+%281%2F8%29%29
Now we just have to simplify the right side. We can use the property of logarithms, log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29, to separate the 1 and 8 of 1/8 into separate logarithms. (You'll see why in a moment.)
x+=+log%282%2C+%281%29%29+-+log%282%2C+%288%29%29
Since any number (except zero) to the zero power is 1, the first logarithm on the right is zero:
x+=+0+-+log%282%2C+%288%29%29
which simplifies to:
x+=+-log%282%2C+%288%29%29
I hope now you can figure out the remaining logarithm. What power of 2 results in 8? Answer: 3. So
x = -3

-3, by the way, was the answer to the original, quick-solution question: "What power of 2 is 1/8?" Look at this for a bit to see if this makes sense now.