log102I

log102I is not a standard mathematical term, and its meaning depends on context. In many cases I is used to denote either the identity matrix or the imaginary unit, or it may be a shorthand in a programming or mathematical text. Because there is no universal definition, readers should rely on surrounding material to determine the intended meaning.

If I denotes the identity matrix of size n, then log_10(2I) is a simple matrix expression: it equals log_10(2) times the identity matrix. This follows from the matrix logarithm identity ln(2I) = (ln 2) I and the base-10 relation log_10(A) = ln(A)/ln(10). The result is an n×n diagonal matrix with every diagonal entry equal to log_10(2). This interpretation keeps the result real and straightforward for all sizes of I.

If I denotes the imaginary unit i, then log_10(2i) is a complex number. Using the principal branch, log_10(2i) = log_10|2i| + i Arg(2i)/ln(10) = log_10(2) + i(pi/2)/ln(10). Numerically, this is approximately 0.3010 + 0.6812 i. The complex logarithm is multivalued, so other values exist of the form log_10(2) + i(pi/2 + 2k pi)/ln(10) for integers k.

In programming or textual notation, log102I might also appear as a misprint or shorthand for log10(2) with an attached symbol I. In any case, restoring precision by writing log_10(2) or log_10(2I) with explicit definitions of I is advisable.