SOLUTION: Solve for x: log_{8}1/256 = x

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Question 409680: Solve for x:
log_{8}1/256 = x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%288%2C+%281%2F256%29%29+=+x
First we'll write this in exponential form. In general log%28a%2C+%28p%29%29+=+q is equivalent to p+=+a%5Eq. Using this pattern on your equation we get:
1%2F256+=+8%5Ex
Since both 8 and 1/256 are powers of 2, 8+=+2%5E3 and 256+=+2%5E8 (so 1%2F256+=+2%5E%28-8%29), we will be able to find the solution "by hand". Rewriting the 8 and the 1/256 as a powers of 2 we get:
2%5E%28-8%29+=+%282%5E3%29%5Ex
On the right side the rule for exponents when raising a power to a power tells use to multiply the exponents:
2%5E%28-8%29+=+2%5E%283%2Ax%29
Now the equation says that two powers of 2 are equal. The only way this can be true is if the exponents themselves are equal, too. So:
-8 = 3x
Dividding by 3 we get:
-8%2F3+=+x