Logb1

logb1 denotes the logarithm of 1 with base b, written as log base b of 1 or log_b(1). It is defined as the exponent y that satisfies b^y = 1.

In the real numbers, this expression is defined only when the base b is positive and not equal to 1. Under these conditions, log_b(1) equals 0, since b^0 = 1 for any such base. This reflects the general property that 1 is the multiplicative identity: raising the base to the power 0 yields 1.

If the base does not meet the real-valued domain requirements (for example, b ≤ 0 or b = 1), log_b(1) is undefined in real arithmetic. In complex analysis, the logarithm is multivalued, and log_b(1) may be represented as Log(1)/Log(b) using the complex logarithm. Since Log(1) can be 2π i k for any integer k, this leads to a set of possible values (2π i k)/Log(b) in general, with the principal value typically taken as 0 when b is a positive real number not equal to 1.

Examples: log_2(1) = 0 and log_10(1) = 0. The result reflects the fact that any positive base raised to the power 0 equals 1.