   SEARCH HOME Math Central Quandaries & Queries  Question from Peter, a parent: How do I solve for "x" in the following equation: "2 to the power of x = 1,000,000" ? Hi Peter,Fthan

I know a few powers of 2, for example 22 = 4, 23 = 8 and so on, but I don't know an x that gives 2x = 1,000,000. Nevertheless I would like to have an idea of the size of x before I try to solve for it.

I do know a couple of facts. In the equation 2x = 1,000,000 if x is a positive integer then 2x is a power of 2 and hence not divisible by 5. But 2x = 1,000,000 and 1,000,000 is divisible by 5. Thus x is not a positive integer.

I also know that 210 = 1,024 so 220 = (210)2 = (1024)2 is slightly larger than 1,0002 = 1,000,000. Thus x is between 10 and 20, and likely much closer to 20 then 10.

To solve 2x = 1,000,000 I would take the log of both sides to get

log(2x) = log(1,000,000)

and from the properties of logs

x log(2) = log(1,000,000)

Solve for x.

I hope this helps,
Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.