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Theorem List for Metamath Proof Explorer - 24101-24200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremefsubm 24101* The image of a subgroup of the group +, under the exponential function of a scaled complex number is a submonoid of the multiplicative group of fld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝐹 = (𝑥𝑋 ↦ (exp‘(𝐴 · 𝑥)))    &   𝐺 = ((mulGrp‘ℂfld) ↾s ran 𝐹)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝑋 ∈ (SubGrp‘ℂfld))       (𝜑 → ran 𝐹 ∈ (SubMnd‘(mulGrp‘ℂfld)))

Theoremcircgrp 24102 The circle group 𝑇 is an Abelian group. (Contributed by Paul Chapman, 25-Mar-2008.) (Revised by Mario Carneiro, 13-May-2014.) (Revised by Thierry Arnoux, 26-Jan-2020.)
𝐶 = (abs “ {1})    &   𝑇 = ((mulGrp‘ℂfld) ↾s 𝐶)       𝑇 ∈ Abel

Theoremcircsubm 24103 The circle group 𝑇 is a submonoid of the multiplicative group of fld. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝐶 = (abs “ {1})    &   𝑇 = ((mulGrp‘ℂfld) ↾s 𝐶)       𝐶 ∈ (SubMnd‘(mulGrp‘ℂfld))

Theoremrzgrp 24104 The quotient group R/Z is a group. (Contributed by Thierry Arnoux, 26-Jan-2020.)
𝑅 = (ℝfld /s (ℝfld ~QG ℤ))       𝑅 ∈ Grp

14.3.4  The natural logarithm on complex numbers

Syntaxclog 24105 Extend class notation with the natural logarithm function on complex numbers.
class log

Syntaxccxp 24106 Extend class notation with the complex power function.
class 𝑐

Definitiondf-log 24107 Define the natural logarithm function on complex numbers. See http://en.wikipedia.org/wiki/Natural_logarithm ("The natural logarithm function can also be defined as the inverse function of the exponential function"). To obtain a function, only the principle value of the multivalued inverses of the exponential function, i.e. the inverse whose imaginary part lies in the interval (-pi, pi], see https://en.wikipedia.org/wiki/Complex_logarithm. (Contributed by Paul Chapman, 21-Apr-2008.)
log = (exp ↾ (ℑ “ (-π(,]π)))

Definitiondf-cxp 24108* Define the power function on complex numbers. Note that the value of this function when 𝑥 = 0 and (ℜ‘𝑦) ≤ 0, 𝑦 ≠ 0 should properly be undefined, but defining it by convention this way simplifies the domain. (Contributed by Mario Carneiro, 2-Aug-2014.)
𝑐 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ if(𝑥 = 0, if(𝑦 = 0, 1, 0), (exp‘(𝑦 · (log‘𝑥)))))

Theoremlogrn 24109 The range of the natural logarithm function, also the principal domain of the exponential function. This allows us to write the longer class expression as simply ran log. (Contributed by Paul Chapman, 21-Apr-2008.) (Revised by Mario Carneiro, 13-May-2014.)
ran log = (ℑ “ (-π(,]π))

Theoremellogrn 24110 Write out the property 𝐴 ∈ ran log explicitly. (Contributed by Mario Carneiro, 1-Apr-2015.)
(𝐴 ∈ ran log ↔ (𝐴 ∈ ℂ ∧ -π < (ℑ‘𝐴) ∧ (ℑ‘𝐴) ≤ π))

Theoremdflog2 24111 The natural logarithm function in terms of the exponential function restricted to its principal domain. (Contributed by Paul Chapman, 21-Apr-2008.)
log = (exp ↾ ran log)

Theoremrelogrn 24112 The range of the natural logarithm function includes the real numbers. (Contributed by Paul Chapman, 21-Apr-2008.) (Proof shortened by Mario Carneiro, 1-Apr-2015.)
(𝐴 ∈ ℝ → 𝐴 ∈ ran log)

Theoremlogrncn 24113 The range of the natural logarithm function is a subset of the complex numbers. (Contributed by Mario Carneiro, 13-May-2014.)
(𝐴 ∈ ran log → 𝐴 ∈ ℂ)

Theoremeff1o2 24114 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 24115 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 24116 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 24117 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 24118 Closure of the natural logarithm function. (Contributed by Paul Chapman, 21-Apr-2008.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (log‘𝐴) ∈ ran log)

Theoremlogcl 24119 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 24120 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 24121 The logarithm of a nonzero complex number is a complex number. Deduction form of logcl 24119. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (log‘𝑋) ∈ ℂ)

Theoremlogimcld 24122 The imaginary part of the logarithm is in (-π(,]π). Deduction form of logimcl 24120. Compare logimclad 24123. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (-π < (ℑ‘(log‘𝑋)) ∧ (ℑ‘(log‘𝑋)) ≤ π))

Theoremlogimclad 24123 The imaginary part of the logarithm is in (-π(,]π). Alternate form of logimcld 24122. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (ℑ‘(log‘𝑋)) ∈ (-π(,]π))

Theoremabslogimle 24124 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 24125 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 24126 Closure of the natural logarithm function on positive reals. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ+ → (log‘𝐴) ∈ ℝ)

Theoremeflog 24127 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 24128 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 24129 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 24130 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 24131 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ+ → (exp‘(log‘𝐴)) = 𝐴)

Theoremlogef 24132 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 24133 Relationship between the natural logarithm function and the exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.)
(𝐴 ∈ ℝ → (log‘(exp‘𝐴)) = 𝐴)

Theoremlogeftb 24134 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 24135 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 24136 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 24137 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 24138 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 24139 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 24140 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 24141 Lemma for relogmul 24142 and relogdiv 24143. 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 24142 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 24143 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 24144 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 24145 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 24146 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 24147 Real part of a logarithm. (Contributed by Mario Carneiro, 15-Sep-2014.)
((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (ℜ‘(log‘𝐴)) = (log‘(abs‘𝐴)))

Theoremrelogiso 24148 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 24149 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 24150 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 24151* 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 24152* 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 24153 Natural logarithm preserves . (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))

Theoremrplogcl 24154 Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
((𝐴 ∈ ℝ ∧ 1 < 𝐴) → (log‘𝐴) ∈ ℝ+)

Theoremlogge0 24155 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 24156 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 24157 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 24158 The cosine of the argument is the quotient of the real part and the absolute value. Compare to efiarg 24157. (Contributed by David Moews, 28-Feb-2017.)
(𝜑𝑋 ∈ ℂ)    &   (𝜑𝑋 ≠ 0)       (𝜑 → (cos‘(ℑ‘(log‘𝑋))) = ((ℜ‘𝑋) / (abs‘𝑋)))

Theoremcosarg0d 24159 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 24160 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 24161 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 24162 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 24163 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 24164 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 24165 The logarithm of the negative of a number with positive imaginary part is i · π less than the original. (Compare logneg 24138.) (Contributed by Mario Carneiro, 3-Apr-2015.)
((𝐴 ∈ ℂ ∧ 0 < (ℑ‘𝐴)) → (log‘-𝐴) = ((log‘𝐴) − (i · π)))

Theoremlogmul2 24166 Generalization of relogmul 24142 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 24167 Generalization of relogdiv 24143 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 24168 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 24169 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 24170 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 24171 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 24172 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 24173 Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)       (𝜑 → (log‘𝐴) ∈ ℝ)

Theoremreeflogd 24174 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 24175 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 24176 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 24177 Natural logarithm preserves . (Contributed by Mario Carneiro, 29-May-2016.)
(𝜑𝐴 ∈ ℝ+)    &   (𝜑𝐵 ∈ ℝ+)       (𝜑 → (𝐴𝐵 ↔ (log‘𝐴) ≤ (log‘𝐵)))

Theoremrelogefd 24178 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 24179 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 24180 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 24181 The inverse logarithm function converges to zero. (Contributed by Mario Carneiro, 30-May-2016.)
(𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0

Theoremlogno1 24182 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 24183 The derivative of the real logarithm function. (Contributed by Mario Carneiro, 24-Feb-2015.)
(ℝ D (log ↾ ℝ+)) = (𝑥 ∈ ℝ+ ↦ (1 / 𝑥))

Theoremrelogcn 24184 The real logarithm function is continuous. (Contributed by Mario Carneiro, 17-Feb-2015.)
(log ↾ ℝ+) ∈ (ℝ+cn→ℝ)

Theoremellogdm 24185 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 24186 A number in the continuous domain of log is nonzero. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝐴𝐷𝐴 ≠ 0)

Theoremlogdmnrp 24187 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 24188 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 24189 Lemma for logcn 24193. (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 24190 Lemma for logcn 24193. (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 24191 Lemma for logcn 24193. (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 24192* Lemma for logcn 24193. (Contributed by Mario Carneiro, 18-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (𝑥𝐷 ↦ (ℑ‘(log‘𝑥))) ∈ (𝐷cn→ℝ)

Theoremlogcn 24193 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 24194 Lemma for dvlog 24197. (Contributed by Mario Carneiro, 24-Feb-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       (log “ 𝐷) ∈ (TopOpen‘ℂfld)

Theoremlogdmopn 24195 The "continuous domain" of log is an open set. (Contributed by Mario Carneiro, 7-Apr-2015.)
𝐷 = (ℂ ∖ (-∞(,]0))       𝐷 ∈ (TopOpen‘ℂfld)

Theoremlogf1o2 24196 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 24197* 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 24198 Lemma for dvlog2 24199. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       𝑆 ⊆ (ℂ ∖ (-∞(,]0))

Theoremdvlog2 24199* 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 24197. (Contributed by Mario Carneiro, 1-Mar-2015.)
𝑆 = (1(ball‘(abs ∘ − ))1)       (ℂ D (log ↾ 𝑆)) = (𝑥𝑆 ↦ (1 / 𝑥))

Theoremadvlog 24200 The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)
(ℝ D (𝑥 ∈ ℝ+ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ+ ↦ (log‘𝑥))

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