Assuming all the logarithms below are taken to the base a (with a, b, c…

Concept

Use the standard logarithm laws (valid whenever every argument is positive): log(xy) = log x + log y (product rule), log(x/y) = log x − log y (quotient rule), and log(xn) = n·log x (power rule). Also, for any valid base a (a > 0, a ≠ 1), loga(a) = 1 — the log of the base to itself is always 1.

Step-by-step simplification

1. Expand log(ab2) = log a + 2 log b, using the product rule and the power rule on b2.

2. Expand log(ac) = log a + log c, using the product rule.

3. Expand log(abc4) = log a + log b + 4 log c, using the product rule and the power rule on c4.

4. Expand 3 log(bc) = 3 log b + 3 log c: first apply the product rule to log(bc) = log b + log c, then scale the result by the coefficient 3.

5. Substitute all four expansions back into the original expression with their original signs, then collect the log a, log b, and log c terms separately: the log a coefficients give 1 − 1 + 1; the log b coefficients give 2 + 1 − 3 = 0; the log c coefficients give −1 + 4 − 3 = 0.

6. Only the log a term survives, so the whole expression reduces exactly to log a. Since a is itself the base of the logarithm here (as stated), log a = loga(a), which equals 1 by the base identity above.

Cross-check

Pick concrete numbers to confirm the cancellation is genuine and not a coincidence of this one case. Let a = 2, b = 3, c = 5 (all positive, a ≠ 1), with the logarithm taken to base a = 2 as stated: the expression becomes log2(18) − log2(10) + log2(3750) − 3 log2(15) = log2[(18 × 3750) / (10 × 153)] = log2(67500 / 33750) = log2(2) = 1 — matching the general result for any choice of positive a, b, c (a ≠ 1) for which the logarithms are defined.

Answer

The expression always simplifies to loga(a), which equals 1.