What is the solution to the equation 1 over the square root of 8 = 4^(m − 2)?

Start by simplifying the 1/sqrt8 on the left side of the equation to get a decimal of .35355=4^(m-2).  Now take the common log of both sides to get
log .35355 = log 4^(m-2).  One of the properties of common logs tells us that we are allowed to bring down the exponent in front which would give us this:
log .35355 = (m-2) log 4.  Divide both sides by log 4 and the do that math on your calculator to get -.75000=m-2.  Add 2 to both sides to solve for m:
2-.75000 = m and m = 1.25