Metamath Proof Explorer


Theorem relogbzexpd

Description: Power law for the general logarithm for integer powers: The logarithm of a positive real number to the power of an integer is equal to the product of the exponent and the logarithm of the base of the power, a deduction version. (Contributed by metakunt, 22-May-2024)

Ref Expression
Hypotheses relogbzexpd.1 φ B +
relogbzexpd.2 φ B 1
relogbzexpd.3 φ C +
relogbzexpd.4 φ N
Assertion relogbzexpd φ log B C N = N log B C

Proof

Step Hyp Ref Expression
1 relogbzexpd.1 φ B +
2 relogbzexpd.2 φ B 1
3 relogbzexpd.3 φ C +
4 relogbzexpd.4 φ N
5 1 rpcnd φ B
6 1 rpne0d φ B 0
7 6 2 nelprd φ ¬ B 0 1
8 5 7 eldifd φ B 0 1
9 8 3 4 3jca φ B 0 1 C + N
10 relogbzexp B 0 1 C + N log B C N = N log B C
11 9 10 syl φ log B C N = N log B C