Metamath Proof Explorer


Theorem rplog11d

Description: The natural logarithm is one-to-one on positive reals. (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses rplog11d.a φ A +
rplog11d.b φ B +
Assertion rplog11d φ log A = log B A = B

Proof

Step Hyp Ref Expression
1 rplog11d.a φ A +
2 rplog11d.b φ B +
3 1 rpcnd φ A
4 2 rpcnd φ B
5 1 rpne0d φ A 0
6 2 rpne0d φ B 0
7 3 4 5 6 log11d φ log A = log B A = B