Proof: Log Product Rule

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	bm = x	if logbx = m, then bm = x (Converse of Log Definition)
2	bn = y	if logby = n, then bn = y (Converse of Log Definition)
3	bo = x ⋅ y	if logb(x ⋅ y) = o, then bo = x ⋅ y (Converse of Log Definition)
4	bo = (bm) ⋅ (bn)	if bm = x and bn = y and bo = x ⋅ y, then bo = (bm) ⋅ (bn) (Multiplication Substitution)
5	(bm) ⋅ (bn) = b(m + n)	(bm) ⋅ (bn) = b(m + n) (power_symmetry)
6	bo = b(m + n)	if bo = (bm) ⋅ (bn) and (bm) ⋅ (bn) = b(m + n), then bo = b(m + n) (Transitive Property of Equality)
7	o = m + n	if bo = b(m + n), then o = m + n (Converse of Power Substitution)
8	logb(x ⋅ y) = (logbx) + (logby)	if logbx = m and logby = n and logb(x ⋅ y) = o and o = m + n, then logb(x ⋅ y) = (logbx) + (logby) (Addition Substitution)

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