If log107 = a, then log10(1/70) is equal to:

Concept:

Two logarithm identities are used here: the reciprocal rule log10(1/x) = -log10x, and the product rule log10(mn) = log10m + log10n. Also, log1010 = 1 by the definition of a base-10 logarithm.

Application:

1. Write 70 as the product 7 × 10.

2. Apply the reciprocal rule: log10(1/70) = -log1070.

3. Apply the product rule: log1070 = log10(7×10) = log107 + log1010.

4. Since log1010 = 1 and log107 = a (given), log1070 = a + 1.

5. Substitute back: log10(1/70) = -(a+1).

Cross-check:

Take a ≈ 0.845 (the actual value of log107). Then log1070 ≈ 1.845, so log10(1/70) ≈ -1.845, which matches -(1+a) ≈ -1.845 — confirming the result independently of the algebra above.

Result:

Hence log10(1/70) = -(1+a).