P. 144 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14301-14400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	logrpap0 14301	The logarithm is apart from 0 if its argument is apart from 1. (Contributed by Jim Kingdon, 5-Jul-2024.)
|- ( ( A e. RR+ /\ A # 1 ) -> ( log ` A ) # 0 )
Theorem	logrpap0d 14302	Deduction form of logrpap0 14301. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( log ` A ) # 0 )
Theorem	rplogcl 14303	Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 21-Sep-2014.)
|- ( ( A e. RR /\ 1 < A ) -> ( log ` A ) e. RR+ )
Theorem	logge0 14304	The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ( A e. RR /\ 1 <_ A ) -> 0 <_ ( log ` A ) )
Theorem	logdivlti 14305	The log x / x function is strictly decreasing on the reals greater than _e. (Contributed by Mario Carneiro, 14-Mar-2014.)
|- ( ( ( A e. RR /\ B e. RR /\ _e <_ A ) /\ A < B ) -> ( ( log ` B ) / B ) < ( ( log ` A ) / A ) )
Theorem	relogcld 14306	Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( log ` A ) e. RR )
Theorem	reeflogd 14307	Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( exp ` ( log ` A ) ) = A )
Theorem	relogmuld 14308	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.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR+ )   =>    |- ( ph -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )
Theorem	relogdivd 14309	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.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR+ )   =>    |- ( ph -> ( log ` ( A / B ) ) = ( ( log ` A ) - ( log ` B ) ) )
Theorem	logled 14310	Natural logarithm preserves <_. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR+ )   =>    |- ( ph -> ( A <_ B <-> ( log ` A ) <_ ( log ` B ) ) )
Theorem	relogefd 14311	Relationship between the natural logarithm function and the exponential function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   =>    |- ( ph -> ( log ` ( exp ` A ) ) = A )
Theorem	rplogcld 14312	Closure of the logarithm function in the positive reals. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 < A )   =>    |- ( ph -> ( log ` A ) e. RR+ )
Theorem	logge0d 14313	The logarithm of a number greater than 1 is nonnegative. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 <_ A )   =>    |- ( ph -> 0 <_ ( log ` A ) )
Theorem	logge0b 14314	The logarithm of a number is nonnegative iff the number is greater than or equal to 1. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( 0 <_ ( log ` A ) <-> 1 <_ A ) )
Theorem	loggt0b 14315	The logarithm of a number is positive iff the number is greater than 1. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( 0 < ( log ` A ) <-> 1 < A ) )
Theorem	logle1b 14316	The logarithm of a number is less than or equal to 1 iff the number is less than or equal to Euler's constant. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( ( log ` A ) <_ 1 <-> A <_ _e ) )
Theorem	loglt1b 14317	The logarithm of a number is less than 1 iff the number is less than Euler's constant. (Contributed by AV, 30-May-2020.)
|- ( A e. RR+ -> ( ( log ` A ) < 1 <-> A < _e ) )
Theorem	rpcxpef 14318	Value of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
Theorem	cxpexprp 14319	Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
Theorem	cxpexpnn 14320	Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. NN /\ B e. ZZ ) -> ( A ^c B ) = ( A ^ B ) )
Theorem	logcxp 14321	Logarithm of a complex power. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR ) -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
Theorem	rpcxp0 14322	Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( A e. RR+ -> ( A ^c 0 ) = 1 )
Theorem	rpcxp1 14323	Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. RR+ -> ( A ^c 1 ) = A )
Theorem	1cxp 14324	Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. CC -> ( 1 ^c A ) = 1 )
Theorem	ecxp 14325	Write the exponential function as an exponent to the power _e. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. CC -> ( _e ^c A ) = ( exp ` A ) )
Theorem	rpcncxpcl 14326	Closure of the complex power function. (Contributed by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) e. CC )
Theorem	rpcxpcl 14327	Positive real closure of the complex power function. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )
Theorem	cxpap0 14328	Complex exponentiation is apart from zero. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 12-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) # 0 )
Theorem	rpcxpadd 14329	Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )
Theorem	rpcxpp1 14330	Value of a nonzero complex number raised to a complex power plus one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B + 1 ) ) = ( ( A ^c B ) x. A ) )
Theorem	rpcxpneg 14331	Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c -u B ) = ( 1 / ( A ^c B ) ) )
Theorem	rpcxpsub 14332	Exponent subtraction law for complex exponentiation. (Contributed by Mario Carneiro, 22-Sep-2014.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B - C ) ) = ( ( A ^c B ) / ( A ^c C ) ) )
Theorem	rpmulcxp 14333	Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A x. B ) ^c C ) = ( ( A ^c C ) x. ( B ^c C ) ) )
Theorem	cxprec 14334	Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
Theorem	rpdivcxp 14335	Complex exponentiation of a quotient. (Contributed by Mario Carneiro, 8-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A / B ) ^c C ) = ( ( A ^c C ) / ( B ^c C ) ) )
Theorem	cxpmul 14336	Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) )
Theorem	rpcxproot 14337	The complex power function allows us to write n-th roots via the idiom A ^c ( 1 / N ). (Contributed by Mario Carneiro, 6-May-2015.)
|- ( ( A e. RR+ /\ N e. NN ) -> ( ( A ^c ( 1 / N ) ) ^ N ) = A )
Theorem	abscxp 14338	Absolute value of a power, when the base is real. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ( A e. RR+ /\ B e. CC ) -> ( abs ` ( A ^c B ) ) = ( A ^c ( Re ` B ) ) )
Theorem	cxplt 14339	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )
Theorem	cxple 14340	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( ( ( A e. RR /\ 1 < A ) /\ ( B e. RR /\ C e. RR ) ) -> ( B <_ C <-> ( A ^c B ) <_ ( A ^c C ) ) )
Theorem	rpcxple2 14341	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 8-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
Theorem	rpcxplt2 14342	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ /\ C e. RR+ ) -> ( A < B <-> ( A ^c C ) < ( B ^c C ) ) )
Theorem	cxplt3 14343	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )
Theorem	cxple3 14344	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016.)
|- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )
Theorem	rpcxpsqrt 14345	The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other n-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.)
|- ( A e. RR+ -> ( A ^c ( 1 / 2 ) ) = ( sqr ` A ) )
Theorem	logsqrt 14346	Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
|- ( A e. RR+ -> ( log ` ( sqr ` A ) ) = ( ( log ` A ) / 2 ) )
Theorem	rpcxp0d 14347	Value of the complex power function when the second argument is zero. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( A ^c 0 ) = 1 )
Theorem	rpcxp1d 14348	Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( A ^c 1 ) = A )
Theorem	1cxpd 14349	Value of the complex power function at one. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. CC )   =>    |- ( ph -> ( 1 ^c A ) = 1 )
Theorem	rpcncxpcld 14350	Closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( A ^c B ) e. CC )
Theorem	cxpltd 14351	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 < A )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B < C <-> ( A ^c B ) < ( A ^c C ) ) )
Theorem	cxpled 14352	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> 1 < A )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B <_ C <-> ( A ^c B ) <_ ( A ^c C ) ) )
Theorem	rpcxpsqrtth 14353	Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11040. (Contributed by AV, 23-Dec-2022.)
|- ( A e. RR+ -> ( ( sqr ` A ) ^c 2 ) = A )
Theorem	cxprecd 14354	Complex exponentiation of a reciprocal. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   =>    |- ( ph -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
Theorem	rpcxpcld 14355	Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( A ^c B ) e. RR+ )
Theorem	logcxpd 14356	Logarithm of a complex power. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   =>    |- ( ph -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
Theorem	cxplt3d 14357	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )
Theorem	cxple3d 14358	Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> A < 1 )   &    |- ( ph -> C e. RR )   =>    |- ( ph -> ( B <_ C <-> ( A ^c C ) <_ ( A ^c B ) ) )
Theorem	cxpmuld 14359	Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. RR )   &    |- ( ph -> C e. CC )   =>    |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) )
Theorem	cxpcom 14360	Commutative law for real exponentiation. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. RR+ /\ B e. RR /\ C e. RR ) -> ( ( A ^c B ) ^c C ) = ( ( A ^c C ) ^c B ) )
Theorem	apcxp2 14361	Apartness and real exponentiation. (Contributed by Jim Kingdon, 10-Jul-2024.)
|- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B # C <-> ( A ^c B ) # ( A ^c C ) ) )
Theorem	rpabscxpbnd 14362	Bound on the absolute value of a complex power. (Contributed by Mario Carneiro, 15-Sep-2014.) (Revised by Jim Kingdon, 19-Jun-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   &    |- ( ph -> 0 < ( Re ` B ) )   &    |- ( ph -> M e. RR )   &    |- ( ph -> ( abs ` A ) <_ M )   =>    |- ( ph -> ( abs ` ( A ^c B ) ) <_ ( ( M ^c ( Re ` B ) ) x. ( exp ` ( ( abs ` B ) x. pi ) ) ) )
Theorem	ltexp2 14363	Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)
|- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ 1 < A ) -> ( M < N <-> ( A ^ M ) < ( A ^ N ) ) )
10.1.4  Logarithms to an arbitrary base Define "log using an arbitrary base" function and then prove some of its properties. As with df-relog 14282 this is for real logarithms rather than complex logarithms. 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): ( B logb X ) where B is the base and X is the argument of the logarithm function. An alternative would be to support the notational form ( ( logb ` B ) ` X ); that looks a little more like traditional notation.
Syntax	clogb 14364	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 14365*	Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as ( B logb X ) for "log base B of X". In the most common traditional notation, base B is a subscript of "log". The definition will only be useful where x is a positive real apart from one and where y is a positive real, so the choice of ( CC \ { 0 , 1 } ) and ( CC \ { 0 } ) is somewhat arbitrary (we adopt the definition used in set.mm). (Contributed by David A. Wheeler, 21-Jan-2017.)
|- logb = ( x e. ( CC \ { 0 , 1 } ) , y e. ( CC \ { 0 } ) |-> ( ( log ` y ) / ( log ` x ) ) )
Theorem	rplogbval 14366	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 Jim Kingdon, 3-Jul-2024.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
Theorem	rplogbcl 14367	General logarithm closure. (Contributed by David A. Wheeler, 17-Jul-2017.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B logb X ) e. RR )
Theorem	rplogbid1 14368	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.)
|- ( ( A e. RR+ /\ A # 1 ) -> ( A logb A ) = 1 )
Theorem	rplogb1 14369	The logarithm of 1 to an arbitrary base B is 0. Property 1(b) of [Cohen4] p. 361. See log1 14290. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. RR+ /\ B # 1 ) -> ( B logb 1 ) = 0 )
Theorem	rpelogb 14370	The general logarithm of a number to the base being Euler's constant is the natural logarithm of the number. Put another way, using _e as the base in logb is the same as log. Definition in [Cohen4] p. 352. (Contributed by David A. Wheeler, 17-Oct-2017.) (Revised by David A. Wheeler and AV, 16-Jun-2020.)
|- ( A e. RR+ -> ( _e logb A ) = ( log ` A ) )
Theorem	rplogbchbase 14371	Change of base for logarithms. Property in [Cohen4] p. 367. (Contributed by AV, 11-Jun-2020.)
|- ( ( ( A e. RR+ /\ A # 1 ) /\ ( B e. RR+ /\ B # 1 ) /\ X e. RR+ ) -> ( A logb X ) = ( ( B logb X ) / ( B logb A ) ) )
Theorem	relogbval 14372	Value of the general logarithm with integer base. (Contributed by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) = ( ( log ` X ) / ( log ` B ) ) )
Theorem	relogbzcl 14373	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.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ ) -> ( B logb X ) e. RR )
Theorem	rplogbreexp 14374	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.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ E e. RR ) -> ( B logb ( C ^c E ) ) = ( E x. ( B logb C ) ) )
Theorem	rplogbzexp 14375	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.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ C e. RR+ /\ N e. ZZ ) -> ( B logb ( C ^ N ) ) = ( N x. ( B logb C ) ) )
Theorem	rprelogbmul 14376	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.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A x. C ) ) = ( ( B logb A ) + ( B logb C ) ) )
Theorem	rprelogbmulexp 14377	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.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ /\ E e. RR ) ) -> ( B logb ( A x. ( C ^c E ) ) ) = ( ( B logb A ) + ( E x. ( B logb C ) ) ) )
Theorem	rprelogbdiv 14378	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.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ ( A e. RR+ /\ C e. RR+ ) ) -> ( B logb ( A / C ) ) = ( ( B logb A ) - ( B logb C ) ) )
Theorem	relogbexpap 14379	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.)
|- ( ( B e. RR+ /\ B # 1 /\ M e. ZZ ) -> ( B logb ( B ^ M ) ) = M )
Theorem	nnlogbexp 14380	Identity law for general logarithm with integer base. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ M e. ZZ ) -> ( B logb ( B ^ M ) ) = M )
Theorem	logbrec 14381	Logarithm of a reciprocal changes sign. Particular case of Property 3 of [Cohen4] p. 361. (Contributed by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ A e. RR+ ) -> ( B logb ( 1 / A ) ) = -u ( B logb A ) )
Theorem	logbleb 14382	The general logarithm function is monotone/increasing. See logleb 14299. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by AV, 31-May-2020.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ /\ Y e. RR+ ) -> ( X <_ Y <-> ( B logb X ) <_ ( B logb Y ) ) )
Theorem	logblt 14383	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 14298. (Contributed by Stefan O'Rear, 19-Oct-2014.) (Revised by Thierry Arnoux, 27-Sep-2017.)
|- ( ( B e. ( ZZ>= ` 2 ) /\ X e. RR+ /\ Y e. RR+ ) -> ( X < Y <-> ( B logb X ) < ( B logb Y ) ) )
Theorem	rplogbcxp 14384	Identity law for the general logarithm for real numbers. (Contributed by AV, 22-May-2020.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR ) -> ( B logb ( B ^c X ) ) = X )
Theorem	rpcxplogb 14385	Identity law for the general logarithm. (Contributed by AV, 22-May-2020.)
|- ( ( B e. RR+ /\ B # 1 /\ X e. RR+ ) -> ( B ^c ( B logb X ) ) = X )
Theorem	relogbcxpbap 14386	The logarithm is the inverse of the exponentiation. Observation in [Cohen4] p. 348. (Contributed by AV, 11-Jun-2020.)
|- ( ( ( B e. RR+ /\ B # 1 ) /\ X e. RR+ /\ Y e. RR ) -> ( ( B logb X ) = Y <-> ( B ^c Y ) = X ) )
Theorem	logbgt0b 14387	The logarithm of a positive real number to a real base greater than 1 is positive iff the number is greater than 1. (Contributed by AV, 29-Dec-2022.)
|- ( ( A e. RR+ /\ ( B e. RR+ /\ 1 < B ) ) -> ( 0 < ( B logb A ) <-> 1 < A ) )
Theorem	logbgcd1irr 14388	The logarithm of an integer greater than 1 to an integer base greater than 1 is not rational if the argument and the base are relatively prime. For example, ( 2 logb 9 ) e. ( RR \ QQ ). (Contributed by AV, 29-Dec-2022.)
|- ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) /\ ( X gcd B ) = 1 ) -> ( B logb X ) e. ( RR \ QQ ) )
Theorem	logbgcd1irraplemexp 14389	Lemma for logbgcd1irrap 14391. Apartness of X ^ N and B ^ M. (Contributed by Jim Kingdon, 11-Jul-2024.)
|- ( ph -> X e. ( ZZ>= ` 2 ) )   &    |- ( ph -> B e. ( ZZ>= ` 2 ) )   &    |- ( ph -> ( X gcd B ) = 1 )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( X ^ N ) # ( B ^ M ) )
Theorem	logbgcd1irraplemap 14390	Lemma for logbgcd1irrap 14391. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.)
|- ( ph -> X e. ( ZZ>= ` 2 ) )   &    |- ( ph -> B e. ( ZZ>= ` 2 ) )   &    |- ( ph -> ( X gcd B ) = 1 )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. NN )   =>    |- ( ph -> ( B logb X ) # ( M / N ) )
Theorem	logbgcd1irrap 14391	The logarithm of an integer greater than 1 to an integer base greater than 1 is irrational (in the sense of being apart from any rational number) if the argument and the base are relatively prime. For example, ( 2 logb 9 ) # Q where Q is rational. (Contributed by AV, 29-Dec-2022.)
|- ( ( ( X e. ( ZZ>= ` 2 ) /\ B e. ( ZZ>= ` 2 ) ) /\ ( ( X gcd B ) = 1 /\ Q e. QQ ) ) -> ( B logb X ) # Q )
Theorem	2logb9irr 14392	Example for logbgcd1irr 14388. The logarithm of nine to base two is not rational. Also see 2logb9irrap 14398 which says that it is irrational (in the sense of being apart from any rational number). (Contributed by AV, 29-Dec-2022.)
|- ( 2 logb 9 ) e. ( RR \ QQ )
Theorem	logbprmirr 14393	The logarithm of a prime to a different prime base is not rational. For example, ( 2 logb 3 ) e. ( RR \ QQ ) (see 2logb3irr 14394). (Contributed by AV, 31-Dec-2022.)
|- ( ( X e. Prime /\ B e. Prime /\ X =/= B ) -> ( B logb X ) e. ( RR \ QQ ) )
Theorem	2logb3irr 14394	Example for logbprmirr 14393. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
|- ( 2 logb 3 ) e. ( RR \ QQ )
Theorem	2logb9irrALT 14395	Alternate proof of 2logb9irr 14392: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( 2 logb 9 ) e. ( RR \ QQ )
Theorem	sqrt2cxp2logb9e3 14396	The square root of two to the power of the logarithm of nine to base two is three. ( sqr ` 2 ) and ( 2 logb 9 ) are not rational (see sqrt2irr0 12164 resp. 2logb9irr 14392), satisfying the statement in 2irrexpq 14397. (Contributed by AV, 29-Dec-2022.)
|- ( ( sqr ` 2 ) ^c ( 2 logb 9 ) ) = 3
Theorem	2irrexpq 14397*	There exist real numbers a and b which are not rational such that ( a ^ b ) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named non-rational numbers ( sqr ` 2 ) and ( 2 logb 9 ), see sqrt2irr0 12164, 2logb9irr 14392 and sqrt2cxp2logb9e3 14396. Therefore, this proof is acceptable/usable in intuitionistic logic. For a theorem which is the same but proves that a and b are irrational (in the sense of being apart from any rational number), see 2irrexpqap 14399. (Contributed by AV, 23-Dec-2022.)
|- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ
Theorem	2logb9irrap 14398	Example for logbgcd1irrap 14391. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.)
|- ( Q e. QQ -> ( 2 logb 9 ) # Q )
Theorem	2irrexpqap 14399*	There exist real numbers a and b which are irrational (in the sense of being apart from any rational number) such that ( a ^ b ) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named irrational numbers ( sqr ` 2 ) and ( 2 logb 9 ), see sqrt2irrap 12180, 2logb9irrap 14398 and sqrt2cxp2logb9e3 14396. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.)
|- E. a e. RR E. b e. RR ( A. p e. QQ a # p /\ A. q e. QQ b # q /\ ( a ^c b ) e. QQ )
10.1.5  Quartic binomial expansion
Theorem	binom4 14400	Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11492, but it turns out to be just as much work to put it into this form after clearing all the sums and calculating binomial coefficients.) (Contributed by Mario Carneiro, 6-May-2015.)
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 4 ) = ( ( ( A ^ 4 ) + ( 4 x. ( ( A ^ 3 ) x. B ) ) ) + ( ( 6 x. ( ( A ^ 2 ) x. ( B ^ 2 ) ) ) + ( ( 4 x. ( A x. ( B ^ 3 ) ) ) + ( B ^ 4 ) ) ) ) )