The value of logabc(a2b2c2) is:

Concept

For any valid base b (b > 0, b ≠ 1), the logarithm logb(x) returns the exponent to which b must be raised to give x. Two identities drive this problem: the power rule logb(xk) = k·logb(x), and the base identity logb(b) = 1.

Application

1. Rewrite the argument as a single power of the base: a2b2c2 = (abc)2.

2. Substitute into the expression: logabc(a2b2c2) = logabc((abc)2).

3. Apply the power rule, bringing the exponent in front: logabc((abc)2) = 2·logabc(abc).

4. Apply the base identity logabc(abc) = 1, giving 2·1 = 2.

Cross-check

Set the base abc = B. The expression is logB(B2). By definition this asks: B to what power gives B2? The answer is 2, confirming the result independently of the power rule.