Question

Using a combination of the rules of logarithms, how can In (4/7x6) be fully expanded so all work to receive full credit

Answer 1

First, a quick clarification: your expression "In (4/7x6)" reads ambiguously. Common interpretations and their full expansions are below. Tell me which one you meant if this doesn't match.

Rules used:
- Product rule: ln(ab) = ln a + ln b
- Quotient rule: ln(a/b) = ln a − ln b
- Power rule: ln(a^k) = k ln a

1) If you meant ln((4/7) · 6):
- Combine the numeric factors: (4/7)·6 = 24/7, so ln((4/7)·6) = ln(24/7).
- Use quotient rule: ln(24/7) = ln 24 − ln 7.
- Factor 24 → 8·3 and use product rule: ln 24 = ln(8·3) = ln 8 + ln 3.
- Use power rule on 8 = 2^3: ln 8 = ln(2^3) = 3 ln 2.
- Final fully expanded form: 3 ln 2 + ln 3 − ln 7.

2) If you meant ln(4/(7·6)) = ln(4/42):
- Simplify: 4/42 = 2/21, so ln(4/(7·6)) = ln(2/21).
- Use quotient rule: ln(2/21) = ln 2 − ln 21.
- Factor 21 = 3·7 and use product rule: ln 21 = ln 3 + ln 7.
- Final fully expanded form: ln 2 − ln 3 − ln 7.

If you meant a different grouping, tell me the intended parentheses and I’ll expand that version.