Logbxc

Log_b(xc) denotes the logarithm of the product xc with base b, often written log_b(xc). It is the real number y such that b^y = xc. For a real-valued logarithm, the base b must be a positive real number not equal to 1, and the argument xc must be positive.

The domain of log_b(xc) in the real numbers is defined by xc > 0, with x and c real numbers. When both x and c are positive, the expression is well defined and the standard logarithm rules apply. If one allows negative x or c, the product xc may be positive as well, but the individual logarithms log_b x and log_b c are not defined in the reals; in that case, the real-valued interpretation of log_b(xc) does not hold, and a complex-valued logarithm would be required.

Key properties include the additive rule log_b(xc) = log_b x + log_b c, valid whenever x > 0 and c > 0. The change of base formula applies: log_b(xc) = ln(xc) / ln(b) for any base b > 0, b ≠ 1. From this, log_b(xc) can also be written as log_b x + log_b c when the two logarithms are real-valued. Other related rules include log_b(1) = 0 and log_b((xc)^k) = k log_b(xc).

Example: with base b = 10, x = 20 and c = 5, log_10(xc) = log_10(100) = 2, which equals log_10 20 + log_10 5.

Log_b(xc) is a standard tool in algebra for simplifying products inside logarithms and is frequently used in sciences and engineering to transform multiplicative relationships into additive ones.

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