Theorem relogexp

Description: The natural logarithm of positive A raised to an integer power. Property 4 of Cohen p. 301-302, restricted to natural logarithms and integer powers N . (Contributed by Steve Rodriguez, 25-Nov-2007)

	Ref	Expression
Assertion	relogexp	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( log ‘ ( 𝐴 ↑ 𝑁 ) ) = ( 𝑁 · ( log ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		relogcl	⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℝ )
2	1	recnd	⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℂ )
3		efexp	⊢ ( ( ( log ‘ 𝐴 ) ∈ ℂ ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) )
4	2 3	sylan	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) )
5		reeflog	⊢ ( 𝐴 ∈ ℝ+ → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 )
6	5	oveq1d	⊢ ( 𝐴 ∈ ℝ+ → ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) = ( 𝐴 ↑ 𝑁 ) )
7	6	adantr	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( ( exp ‘ ( log ‘ 𝐴 ) ) ↑ 𝑁 ) = ( 𝐴 ↑ 𝑁 ) )
8	4 7	eqtrd	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) = ( 𝐴 ↑ 𝑁 ) )
9	8	fveq2d	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( log ‘ ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) ) = ( log ‘ ( 𝐴 ↑ 𝑁 ) ) )
10		zre	⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ )
11		remulcl	⊢ ( ( 𝑁 ∈ ℝ ∧ ( log ‘ 𝐴 ) ∈ ℝ ) → ( 𝑁 · ( log ‘ 𝐴 ) ) ∈ ℝ )
12	10 1 11	syl2anr	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 · ( log ‘ 𝐴 ) ) ∈ ℝ )
13		relogef	⊢ ( ( 𝑁 · ( log ‘ 𝐴 ) ) ∈ ℝ → ( log ‘ ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) ) = ( 𝑁 · ( log ‘ 𝐴 ) ) )
14	12 13	syl	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( log ‘ ( exp ‘ ( 𝑁 · ( log ‘ 𝐴 ) ) ) ) = ( 𝑁 · ( log ‘ 𝐴 ) ) )
15	9 14	eqtr3d	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑁 ∈ ℤ ) → ( log ‘ ( 𝐴 ↑ 𝑁 ) ) = ( 𝑁 · ( log ‘ 𝐴 ) ) )