What are the powers of 3 in the range 3-1000?

Answer 1

I guess you mean the value of the power of 3 is at most 1000, not the value of 3^3, 3^4, ..., 3^1000

Logarithms tell you the power to which the base must be raised to get the number. eg for common logs (logs base 10) lg 100 = 2 since 10^2 = 100. To find the largest power of 3 which is at most 1000, the logs to base 3 or 1000 needs to be taken; if the result is not a whole number, the decimal portion is ignored (ie the number is truncated) and only the whole number is considered.

[In the following log_3 means logs to base 3, and log is the log to any base, often common logs (lg) or natural logs (ln = logs to base e) are used (for the whole calculation).]

log_3(1000) = log(1000)/log(3) ≈ 6.3 → highest power of 3 less than or equal to 1000 is 6, ie 3^6.

Thus the powers of 3 in the range 3-1000 are:

3^1 = 3

3^2 = 9

3^3 = 27

3^4 = 81

3^5 = 243

3^6 = 729