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4 = 2²

→ 4^x = 2^(2x)

We want 4^x > w

Taking logs gives:

x log 4 > log w

→ x > log w / log 4

This can be solved to find a value of x by using logs to any base (the same base for each log).

Note that as w become less positive the value of x decreases at a much faster rate.

As w tends towards 0+, x will become more and more negative.

If w is negative, log w does not exist and all values of x will solve the problem - this is obvious as 4 (and 2) to all powers are positive which are all greater than a negative number.

If you require an integer solution to the problem, then the set of solutions starts with the first integer that is greater than x and continues with each next integer higher.


  • w = 40
The above gives x ≈ 2.661 → solution set of integers is {3, 4, 5, ...) with corresponding values {64, 256, 1024, ...}
  • w = 64
The above gives x = 3 → solution set of integers is {4, 5, 6, ...} with corresponding values {256, 1024, 4096, ...}
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Q: What example is there for when a power of 2 is equal to a power of 4 and greater than the number W?
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