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Theorem List for Metamath Proof Explorer - 24001-24100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeff1o2 24001 The exponential function restricted to its principal domain maps one-to-one onto the nonzero complex numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
(exp ↾ ran log):ran log–1-1-onto→(ℂ ∖ {0})
 
Theoremlogf1o 24002 The natural logarithm function maps the nonzero complex numbers one-to-one onto its range. (Contributed by Paul Chapman, 21-Apr-2008.)
log:(ℂ ∖ {0})–1-1-onto→ran log
 
Theoremdfrelog 24003 The natural logarithm function on the positive reals in terms of the real exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
(log ↾ ℝ+) = (exp ↾ ℝ)
 
Theoremrelogf1o 24004 The natural logarithm function maps the positive reals one-to-one onto the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.)
(log ↾ ℝ+):ℝ+1-1-onto→ℝ
 
Theoremlogrncl 24005 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ran log)
 
Theoremlogcl 24006 Closure of the natural logarithm function. (Contributed by NM, 21-Apr-2008.) (Revised by Mario Carneiro, 23-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ℂ)
 
Theoremlogimcl 24007 Closure of the imaginary part of the logarithm function. (Contributed by Mario Carneiro, 23-Sep-2014.) (Revised by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-π < (ℑ‘(log‘𝐴)) ∧ (ℑ‘(log‘𝐴)) ≤ π))
 
Theoremlogcld 24008 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 24006. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (log‘𝑋) ∈ ℂ)
 
Theoremlogimcld 24009 The imaginary part of the logarithm is in (-π(,]π). Deduction form of logimcl 24007. Compare logimclad 24010. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (-π < (ℑ‘(log‘𝑋)) ∧ (ℑ‘(log‘𝑋)) ≤ π))
 
Theoremlogimclad 24010 The imaginary part of the logarithm is in (-π(,]π). Alternate form of logimcld 24009. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (ℑ‘(log‘𝑋)) ∈ (-π(,]π))
 
Theoremabslogimle 24011 The imaginary part of the logarithm function has absolute value less than pi. (Contributed by Mario Carneiro, 3-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(ℑ‘(log‘𝐴))) ≤ π)
 
Theoremlogrnaddcl 24012 The range of the natural logarithm is closed under addition with reals. (Contributed by Mario Carneiro, 3-Apr-2015.)
((𝐴 ∈ ran log ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ran log)
 
Theoremrelogcl 24013 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)
 
Theoremeflog 24014 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
 
Theoremlogeq0im1 24015 If the logarithm of a number is 0, the number must be 1. (Contributed by David A. Wheeler, 22-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ (log‘𝐴) = 0) → 𝐴 = 1)
 
Theoremlogccne0 24016 The logarithm isn't 0 if its argument isn't 0 or 1. (Contributed by David A. Wheeler, 17-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐴 ≠ 1) → (log‘𝐴) ≠ 0)
 
Theoremlogne0 24017 Logarithm of a non-1 positive real number is not zero and thus suitable as a divisor. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 14-Jun-2020.)
((𝐴 ∈ ℝ+𝐴 ≠ 1) → (log‘𝐴) ≠ 0)
 
Theoremreeflog 24018 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴)
 
Theoremlogef 24019 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
(𝐴 ∈ ran log → (log‘(exp‘𝐴)) = 𝐴)
 
Theoremrelogef 24020 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ → (log‘(exp‘𝐴)) = 𝐴)
 
Theoremlogeftb 24021 Relationship between the natural logarithm function and the exponential function. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ran log) → ((log‘𝐴) = 𝐵 ↔ (exp‘𝐵) = 𝐴))
 
Theoremrelogeftb 24022 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ) → ((log‘𝐴) = 𝐵 ↔ (exp‘𝐵) = 𝐴))
 
Theoremlog1 24023 The natural logarithm of 1. One case of Property 1a of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(log‘1) = 0
 
Theoremloge 24024 The natural logarithm of e. One case of Property 1b of [Cohen] p. 301. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(log‘e) = 1
 
Theoremlogneg 24025 The natural logarithm of a negative real number. (Contributed by Mario Carneiro, 13-May-2014.) (Revised by Mario Carneiro, 3-Apr-2015.)
(𝐴 ∈ ℝ+ → (log‘-𝐴) = ((log‘𝐴) + (i · π)))
 
Theoremlogm1 24026 The natural logarithm of negative 1. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
(log‘-1) = (i · π)
 
Theoremlognegb 24027 If a number has imaginary part equal to π, then it is on the negative real axis and vice-versa. (Contributed by Mario Carneiro, 23-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (-𝐴 ∈ ℝ+ ↔ (ℑ‘(log‘𝐴)) = π))
 
Theoremrelogoprlem 24028 Lemma for relogmul 24029 and relogdiv 24030. Remark of [Cohen] p. 301 ("The proof of Property 3 is quite similar to the proof given for Property 2"). (Contributed by Steve Rodriguez, 25-Nov-2007.)
(((log‘𝐴) ∈ ℂ ∧ (log‘𝐵) ∈ ℂ) → (exp‘((log‘𝐴)𝐹(log‘𝐵))) = ((exp‘(log‘𝐴))𝐺(exp‘(log‘𝐵))))    &   (((log‘𝐴) ∈ ℝ ∧ (log‘𝐵) ∈ ℝ) → ((log‘𝐴)𝐹(log‘𝐵)) ∈ ℝ)       ((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (log‘(𝐴𝐺𝐵)) = ((log‘𝐴)𝐹(log‘𝐵)))
 
Theoremrelogmul 24029 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))
 
Theoremrelogdiv 24030 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵)))
 
Theoremexplog 24031 Exponentiation of a nonzero complex number to an integer power. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴𝑁) = (exp‘(𝑁 · (log‘𝐴))))
 
Theoremreexplog 24032 Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝑁 ∈ ℤ) → (𝐴𝑁) = (exp‘(𝑁 · (log‘𝐴))))
 
Theoremrelogexp 24033 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.)
((𝐴 ∈ ℝ+𝑁 ∈ ℤ) → (log‘(𝐴𝑁)) = (𝑁 · (log‘𝐴)))
 
Theoremrelog 24034 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))
 
Theoremrelogiso 24035 The natural logarithm function on positive reals determines an isomorphism from the positive reals onto the reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(log ↾ ℝ+) Isom < , < (ℝ+, ℝ)
 
Theoremreloggim 24036 The natural logarithm is a group isomorphism from the group of positive reals under multiplication to the group of reals under addition. (Contributed by Mario Carneiro, 21-Jun-2015.) (Revised by Thierry Arnoux, 30-Jun-2019.)
𝑃 = ((mulGrp‘ℂfld) ↾s+)       (log ↾ ℝ+) ∈ (𝑃 GrpIso ℝfld)
 
Theoremlogltb 24037 The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴 < 𝐵 ↔ (log‘𝐴) < (log‘𝐵)))
 
Theoremlogfac 24038* The logarithm of a factorial can be expressed as a finite sum of logs. (Contributed by Mario Carneiro, 17-Apr-2015.)
(𝑁 ∈ ℕ0 → (log‘(!‘𝑁)) = Σ𝑘 ∈ (1...𝑁)(log‘𝑘))
 
Theoremeflogeq 24039* Solve an equation involving an exponential. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((exp‘𝐴) = 𝐵 ↔ ∃𝑛 ∈ ℤ 𝐴 = ((log‘𝐵) + ((i · (2 · π)) · 𝑛))))
 
Theoremlogleb 24040 Natural logarithm preserves . (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))
 
Theoremrplogcl 24041 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+)
 
Theoremlogge0 24042 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → 0 ≤ (log‘𝐴))
 
Theoremlogcj 24043 The natural logarithm distributes under conjugation away from the branch cut. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) ≠ 0) → (log‘(∗‘𝐴)) = (∗‘(log‘𝐴)))
 
Theoremefiarg 24044 The exponential of the "arg" function ℑ ∘ log. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(i · (ℑ‘(log‘𝐴)))) = (𝐴 / (abs‘𝐴)))
 
Theoremcosargd 24045 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 24044. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋)))
 
Theoremcosarg0d 24046 The cosine of the argument is zero precisely on the imaginary axis. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → ((cos‘(ℑ‘(log‘𝑋))) = 0 ↔ (ℜ‘𝑋) = 0))
 
Theoremargregt0 24047 Closure of the argument of a complex number with positive real part. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ 0 < (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)(,)(π / 2)))
 
Theoremargrege0 24048 Closure of the argument of a complex number with nonnegative real part. (Contributed by Mario Carneiro, 2-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (-(π / 2)[,](π / 2)))
 
Theoremargimgt0 24049 Closure of the argument of a complex number with positive imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (ℑ‘(log‘𝐴)) ∈ (0(,)π))
 
Theoremargimlt0 24050 Closure of the argument of a complex number with negative imaginary part. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ (ℑ‘𝐴) < 0) → (ℑ‘(log‘𝐴)) ∈ (-π(,)0))
 
Theoremlogimul 24051 Multiplying a number by i increases the logarithm of the number by iπ / 2. (Contributed by Mario Carneiro, 4-Apr-2015.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 0 ≤ (ℜ‘𝐴)) → (log‘(i · 𝐴)) = ((log‘𝐴) + (i · (π / 2))))
 
Theoremlogneg2 24052 The logarithm of the negative of a number with positive imaginary part is i · π less than the original. (Compare logneg 24025.) (Contributed by Mario Carneiro, 3-Apr-2015.)
((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘-𝐴) = ((log‘𝐴) − (i · π)))
 
Theoremlogmul2 24053 Generalization of relogmul 24029 to a complex left argument. (Contributed by Mario Carneiro, 9-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))
 
Theoremlogdiv2 24054 Generalization of relogdiv 24030 to a complex left argument. (Contributed by Mario Carneiro, 8-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℝ+) → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵)))
 
Theoremabslogle 24055 Bound on the magnitude of the complex logarithm function. (Contributed by Mario Carneiro, 3-Jul-2017.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘(log‘𝐴)) ≤ ((abs‘(log‘(abs‘𝐴))) + π))
 
Theoremtanarg 24056 The basic relation between the "arg" function ℑ ∘ log and the arctangent. (Contributed by Mario Carneiro, 25-Feb-2015.)
((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≠ 0) → (tan‘(ℑ‘(log‘𝐴))) = ((ℑ‘𝐴) / (ℜ‘𝐴)))
 
Theoremlogdivlti 24057 The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 14-Mar-2014.)
(((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ e ≤ 𝐴) ∧ 𝐴 < 𝐵) → ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴))
 
Theoremlogdivlt 24058 The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 14-Mar-2014.)
(((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴 < 𝐵 ↔ ((log‘𝐵) / 𝐵) < ((log‘𝐴) / 𝐴)))
 
Theoremlogdivle 24059 The log𝑥 / 𝑥 function is strictly decreasing on the reals greater than e. (Contributed by Mario Carneiro, 3-May-2016.)
(((𝐴 ∈ ℝ ∧ e ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ e ≤ 𝐵)) → (𝐴𝐵 ↔ ((log‘𝐵) / 𝐵) ≤ ((log‘𝐴) / 𝐴)))
 
Theoremrelogcld 24060 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (log‘𝐴) ∈ ℝ)
 
Theoremreeflogd 24061 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (exp‘(log‘𝐴)) = 𝐴)
 
Theoremrelogmuld 24062 The natural logarithm of the product of two positive real numbers is the sum of natural logarithms. Property 2 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (log‘(𝐴 · 𝐵)) = ((log‘𝐴) + (log‘𝐵)))
 
Theoremrelogdivd 24063 The natural logarithm of the quotient of two positive real numbers is the difference of natural logarithms. Exercise 72(a) and Property 3 of [Cohen] p. 301, restricted to natural logarithms. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (log‘(𝐴 / 𝐵)) = ((log‘𝐴) − (log‘𝐵)))
 
Theoremlogled 24064 Natural logarithm preserves . (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))
 
Theoremrelogefd 24065 Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)       (𝜑 → (log‘(exp‘𝐴)) = 𝐴)
 
Theoremrplogcld 24066 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 < 𝐴)       (𝜑 → (log‘𝐴) ∈ ℝ+)
 
Theoremlogge0d 24067 The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑 → 1 ≤ 𝐴)       (𝜑 → 0 ≤ (log‘𝐴))
 
Theoremdivlogrlim 24068 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
 
Theoremlogno1 24069 The logarithm function is not eventually bounded. (Contributed by Mario Carneiro, 30-Apr-2016.) (Proof shortened by Mario Carneiro, 30-May-2016.)
¬ (𝑥 ∈ ℝ+ ↦ (log‘𝑥)) ∈ 𝑂(1)
 
Theoremdvrelog 24070 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
(ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))
 
Theoremrelogcn 24071 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
(log ↾ ℝ+) ∈ (ℝ+cn→ℝ)
 
Theoremellogdm 24072 Elementhood in the "continuous domain" of the complex logarithm. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷 ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)))
 
Theoremlogdmn0 24073 A number in the continuous domain of log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷𝐴 ≠ 0)
 
Theoremlogdmnrp 24074 A number in the continuous domain of log is not a strictly negative number. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷 → ¬ -𝐴 ∈ ℝ+)
 
Theoremlogdmss 24075 The continuity domain of log is a subset of the regular domain of log. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       𝐷 ⊆ (ℂ ∖ {0})
 
Theoremlogcnlem2 24076 Lemma for logcn 24080. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴)))    &   𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))    &   (𝜑𝐴𝐷)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → if(𝑆𝑇, 𝑆, 𝑇) ∈ ℝ+)
 
Theoremlogcnlem3 24077 Lemma for logcn 24080. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴)))    &   𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))    &   (𝜑𝐴𝐷)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐷)    &   (𝜑 → (abs‘(𝐴𝐵)) < if(𝑆𝑇, 𝑆, 𝑇))       (𝜑 → (-π < ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ∧ ((ℑ‘(log‘𝐵)) − (ℑ‘(log‘𝐴))) ≤ π))
 
Theoremlogcnlem4 24078 Lemma for logcn 24080. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))    &   𝑆 = if(𝐴 ∈ ℝ+, 𝐴, (abs‘(ℑ‘𝐴)))    &   𝑇 = ((abs‘𝐴) · (𝑅 / (1 + 𝑅)))    &   (𝜑𝐴𝐷)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐷)    &   (𝜑 → (abs‘(𝐴𝐵)) < if(𝑆𝑇, 𝑆, 𝑇))       (𝜑 → (abs‘((ℑ‘(log‘𝐴)) − (ℑ‘(log‘𝐵)))) < 𝑅)
 
Theoremlogcnlem5 24079* Lemma for logcn 24080. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝑥𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷cn→ℝ)
 
Theoremlogcn 24080 The logarithm function is continuous away from the branch cut at negative reals. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log ↾ 𝐷) ∈ (𝐷cn→ℂ)
 
Theoremdvloglem 24081 Lemma for dvlog 24084. (Contributed by Mario Carneiro, 24-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log “ 𝐷) ∈ (TopOpen‘ℂfld)
 
Theoremlogdmopn 24082 The "continuous domain" of log is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       𝐷 ∈ (TopOpen‘ℂfld)
 
Theoremlogf1o2 24083 The logarithm maps its continuous domain bijectively onto the set of numbers with imaginary part -π < ℑ(𝑧) < π. The negative reals are mapped to the numbers with imaginary part equal to π. (Contributed by Mario Carneiro, 2-May-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log ↾ 𝐷):𝐷1-1-onto→(ℑ “ (-π(,)π))
 
Theoremdvlog 24084* The derivative of the complex logarithm function. (Contributed by Mario Carneiro, 25-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (ℂ D (log ↾ 𝐷)) = (𝑥𝐷 ↦ (1 / 𝑥))
 
Theoremdvlog2lem 24085 Lemma for dvlog2 24086. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       𝑆 ⊆ (ℂ ∖ (-∞(,]0))
 
Theoremdvlog2 24086* The derivative of the complex logarithm function on the open unit ball centered at 1, a sometimes easier region to work with than the ℂ ∖ (-∞, 0] of dvlog 24084. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       (ℂ D (log ↾ 𝑆)) = (𝑥𝑆 ↦ (1 / 𝑥))
 
Theoremadvlog 24087 The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)
(ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))
 
Theoremadvlogexp 24088* The antiderivative of a power of the logarithm. (Set 𝐴 = 1 and multiply by (-1)↑𝑁 · 𝑁! to get the antiderivative of log(𝑥)↑𝑁 itself.) (Contributed by Mario Carneiro, 22-May-2016.)
((𝐴 ∈ ℝ+𝑁 ∈ ℕ0) → (ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝐴 / 𝑥))↑𝑘) / (!‘𝑘))))) = (𝑥 ∈ ℝ+ ↦ (((log‘(𝐴 / 𝑥))↑𝑁) / (!‘𝑁))))
 
Theoremefopnlem1 24089 Lemma for efopn 24091. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)
(((𝑅 ∈ ℝ+𝑅 < π) ∧ 𝐴 ∈ (0(ball‘(abs ∘ − ))𝑅)) → (abs‘(ℑ‘𝐴)) < π)
 
Theoremefopnlem2 24090 Lemma for efopn 24091. (Contributed by Mario Carneiro, 2-May-2015.)
𝐽 = (TopOpen‘ℂfld)       ((𝑅 ∈ ℝ+𝑅 < π) → (exp “ (0(ball‘(abs ∘ − ))𝑅)) ∈ 𝐽)
 
Theoremefopn 24091 The exponential map is an open map. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐽 = (TopOpen‘ℂfld)       (𝑆𝐽 → (exp “ 𝑆) ∈ 𝐽)
 
Theoremlogtayllem 24092* Lemma for logtayl 24093. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝑛 = 0, 0, (1 / 𝑛)) · (𝐴𝑛)))) ∈ dom ⇝ )
 
Theoremlogtayl 24093* The Taylor series for -log(1 − 𝐴). (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → seq1( + , (𝑘 ∈ ℕ ↦ ((𝐴𝑘) / 𝑘))) ⇝ -(log‘(1 − 𝐴)))
 
Theoremlogtaylsum 24094* The Taylor series for -log(1 − 𝐴), as an infinite sum. (Contributed by Mario Carneiro, 31-Mar-2015.)
((𝐴 ∈ ℂ ∧ (abs‘𝐴) < 1) → Σ𝑘 ∈ ℕ ((𝐴𝑘) / 𝑘) = -(log‘(1 − 𝐴)))
 
Theoremlogtayl2 24095* Power series expression for the logarithm. (Contributed by Mario Carneiro, 31-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       (𝐴𝑆 → seq1( + , (𝑘 ∈ ℕ ↦ (((-1↑(𝑘 − 1)) / 𝑘) · ((𝐴 − 1)↑𝑘)))) ⇝ (log‘𝐴))
 
Theoremlogccv 24096 The natural logarithm function on the reals is a strictly concave function. (Contributed by Mario Carneiro, 20-Jun-2015.)
(((𝐴 ∈ ℝ+𝐵 ∈ ℝ+𝐴 < 𝐵) ∧ 𝑇 ∈ (0(,)1)) → ((𝑇 · (log‘𝐴)) + ((1 − 𝑇) · (log‘𝐵))) < (log‘((𝑇 · 𝐴) + ((1 − 𝑇) · 𝐵))))
 
Theoremcxpval 24097 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) = if(𝐴 = 0, if(𝐵 = 0, 1, 0), (exp‘(𝐵 · (log‘𝐴)))))
 
Theoremcxpef 24098 Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ) → (𝐴𝑐𝐵) = (exp‘(𝐵 · (log‘𝐴))))
 
Theorem0cxp 24099 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (0↑𝑐𝐴) = 0)
 
Theoremcxpexpz 24100 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℤ) → (𝐴𝑐𝐵) = (𝐴𝐵))
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