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Theorem List for Metamath Proof Explorer - 23701-23800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcxpne0 23701 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpeq0 23702 Complex exponentiation is zero iff the mantissa is zero and the exponent is nonzero. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremcxpadd 23703 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpp1 23704 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpneg 23705 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpsub 23706 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremcxpge0 23707 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremmulcxplem 23708 Lemma for mulcxp 23709. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremmulcxp 23709 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxprec 23710 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremdivcxp 23711 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremcxpmul 23712 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpmul2 23713 Product of exponents law for complex exponentiation. Variation on cxpmul 23712 with more general conditions on and when is an integer. (Contributed by Mario Carneiro, 9-Aug-2014.)

Theoremcxproot 23714 The complex power function allows us to write n-th roots via the idiom . (Contributed by Mario Carneiro, 6-May-2015.)

Theoremcxpmul2z 23715 Generalize cxpmul2 23713 to negative integers. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremabscxp 23716 Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremabscxp2 23717 Absolute value of a power, when the exponent is real. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxplt 23718 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxple 23719 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxplea 23720 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremcxple2 23721 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremcxplt2 23722 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxple2a 23723 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremcxplt3 23724 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxple3 23725 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxpsqrtlem 23726 Lemma for cxpsqrt 23727. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremcxpsqrt 23727 The complex exponential function with exponent exactly matches the complex square root function (the branch cut is in the same place for both functions), and thus serves as a suitable generalization to other -th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.)

Theoremlogsqrt 23728 Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)

Theoremcxp0d 23729 Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxp1d 23730 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)

Theorem1cxpd 23731 Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpcld 23732 Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmul2d 23733 Product of exponents law for complex exponentiation. Variation on cxpmul 23712 with more general conditions on and when is an integer. (Contributed by Mario Carneiro, 30-May-2016.)

Theorem0cxpd 23734 Value of the complex power function when the first argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpexpzd 23735 Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpefd 23736 Value of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpne0d 23737 Complex exponentiation is nonzero if its mantissa is nonzero. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpp1d 23738 Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpnegd 23739 Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmul2zd 23740 Generalize cxpmul2 23713 to negative integers. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpaddd 23741 Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpsubd 23742 Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpltd 23743 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpled 23744 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplead 23745 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremdivcxpd 23746 Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremrecxpcld 23747 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpge0d 23748 Nonnegative exponentiation with a real exponent is nonnegative. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple2ad 23749 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplt2d 23750 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple2d 23751 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremmulcxpd 23752 Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxprecd 23753 Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremrpcxpcld 23754 Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremlogcxpd 23755 Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxplt3d 23756 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxple3d 23757 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremcxpmuld 23758 Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)

Theoremdvcxp1 23759* The derivative of a complex power with respect to the first argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvcxp2 23760* The derivative of a complex power with respect to the second argument. (Contributed by Mario Carneiro, 24-Feb-2015.)

Theoremdvsqrt 23761 The derivative of the real square root function. (Contributed by Mario Carneiro, 1-May-2016.)

Theoremdvcncxp1 23762* Derivative of complex power with respect to first argument on the complex plane. (Contributed by Brendan Leahy, 18-Dec-2018.)

Theoremdvcnsqrt 23763* Derivative of square root function. (Contributed by Brendan Leahy, 18-Dec-2018.)

Theoremcxpcn 23764* Domain of continuity of the complex power function. (Contributed by Mario Carneiro, 1-May-2016.)
fld       t

Theoremcxpcn2 23765* Continuity of the complex power function, when the base is real. (Contributed by Mario Carneiro, 1-May-2016.)
fld       t

Theoremcxpcn3lem 23766* Lemma for cxpcn3 23767. (Contributed by Mario Carneiro, 2-May-2016.)
fld       t        t

Theoremcxpcn3 23767* Extend continuity of the complex power function to a base of zero, as long as the exponent has strictly positive real part. (Contributed by Mario Carneiro, 2-May-2016.)
fld       t        t

Theoremresqrtcn 23768 Continuity of the real square root function. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremsqrtcn 23769 Continuity of the square root function. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremcxpaddle 23771 Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremabscxpbnd 23772 Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.)

Theoremroot1id 23773 Property of an -th root of unity. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremroot1eq1 23774 The only powers of an -th root of unity that equal are the multiples of . In other words, has order in the multiplicative group of nonzero complex numbers. (In fact, these and their powers are the only elements of finite order in the complex numbers.) (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremroot1cj 23775 Within the -th roots of unity, the conjugate of the -th root is the -th root. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremcxpeq 23776* Solve an equation involving an -th power. The expression is a way to write the primitive -th root of unity with the smallest positive argument. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremloglesqrt 23777 An upper bound on the logarithm. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremlogreclem 23778 Symmetry of the natural logarithm range by negation. Lemma for logrec 23779. (Contributed by Saveliy Skresanov, 27-Dec-2016.)

Theoremlogrec 23779 Logarithm of a reciprocal changes sign. (Contributed by Saveliy Skresanov, 28-Dec-2016.)

14.3.5  Logarithms to an arbitrary base

Define "log using an arbitrary base" function and then prove some of its properties. Note that logb is generalized to an arbitrary base and arbitrary parameter in , but it doesn't accept infinities as arguments, unlike .

Metamath doesn't care what letters are used to represent classes. Usually classes begin with the letter "A", but here we use "B" and "X" to more clearly distinguish between "base" and "other parameter of log".

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions (operations): logb where is the base and is the argument of the logarithm function. An alternative would be to support the notational form logb ; that looks a little more like traditional notation. Such a function logb for a fixed base can be obtained in Metamath (without the need for a new definition) by the curry function: curry logb , see logbmpt 23804, logbf 23805 and logbfval 23806.

Syntaxclogb 23780 Extend class notation to include the logarithm generalized to an arbitrary base.
logb

Definitiondf-logb 23781* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as logb for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition is according to Wikipedia "Complex logarithm": https://en.wikipedia.org/wiki/Complex_logarithm#Logarithms_to_other_bases (10-Jun-2020). (Contributed by David A. Wheeler, 21-Jan-2017.)
logb

Theoremlogbval 23782 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the argument of the logarithm function here. (Contributed by David A. Wheeler, 21-Jan-2017.) (Revised by David A. Wheeler, 16-Jul-2017.)
logb

Theoremlogbcl 23783 General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
logb

Theoremlogbid1 23784 General logarithm is 1 when base and arg match. Property 1(a) of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by David A. Wheeler, 22-Jul-2017.)
logb

Theoremlogb1 23785 The logarithm of to an arbitrary base is 0. Property 1(b) of [Cohen4] p. 361. See log1 23614. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremelogb 23786 The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using as the base in logb is the same as . Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.)
logb

Theoremlogbchbase 23787 Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
logb logb logb

Theoremrelogbval 23788 Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremrelogbcl 23789 Closure of the general logarithm with a positive real base on positive reals. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremrelogbzcl 23790 Closure of the general logarithm with integer base on positive reals. (Contributed by Thierry Arnoux, 27-Sep-2017.) (Proof shortened by AV, 9-Jun-2020.)
logb

Theoremrelogbreexp 23791 Power law for the general logarithm for real powers: The logarithm of a positive real number to the power of a real number is equal to the product of the exponent and the logarithm of the base of the power. Property 4 of [Cohen4] p. 361. (Contributed by AV, 9-Jun-2020.)
logb logb

Theoremrelogbzexp 23792 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. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
logb logb

Theoremrelogbmul 23793 The logarithm of the product of two positive real numbers is the sum of logarithms. Property 2 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 29-May-2020.)
logb logb logb

Theoremrelogbmulexp 23794 The logarithm of the product of a positive real and a positive real number to the power of a real number is the sum of the logarithm of the first real number and the scaled logarithm of the second real number. (Contributed by AV, 29-May-2020.)
logb logb logb

Theoremrelogbdiv 23795 The logarithm of the quotient of two positive real numbers is the difference of logarithms. Property 3 of [Cohen4] p. 361. (Contributed by AV, 29-May-2020.)
logb logb logb

Theoremrelogbexp 23796 Identity law for general logarithm: the logarithm of a power to the base is the exponent. Property 6 of [Cohen4] p. 361. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by AV, 9-Jun-2020.)
logb

Theoremnnlogbexp 23797 Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
logb

Theoremlogbrec 23798 Logarithm of a reciprocal changes sign. See logrec 23779. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.)
logb logb

Theoremlogbleb 23799 The general logarithm function is monotone/increasing. See logleb 23631. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
logb logb

Theoremlogblt 23800 The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 23628. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
logb logb

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