P. 157 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15601-15700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	logltb 15601	The natural logarithm function on positive reals is strictly monotonic. (Contributed by Steve Rodriguez, 25-Nov-2007.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A < B <-> ( log ` A ) < ( log ` B ) ) )
Theorem	logleb 15602	Natural logarithm preserves <_. (Contributed by Stefan O'Rear, 19-Sep-2014.)
|- ( ( A e. RR+ /\ B e. RR+ ) -> ( A <_ B <-> ( log ` A ) <_ ( log ` B ) ) )
Theorem	logrpap0b 15603	The logarithm is apart from 0 if and only if its argument is apart from 1. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( A e. RR+ -> ( A # 1 <-> ( log ` A ) # 0 ) )
Theorem	logrpap0 15604	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 15605	Deduction form of logrpap0 15604. (Contributed by Jim Kingdon, 3-Jul-2024.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> A # 1 )   =>    |- ( ph -> ( log ` A ) # 0 )
Theorem	rplogcl 15606	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 15607	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 15608	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 15609	Closure of the natural logarithm function. (Contributed by Mario Carneiro, 29-May-2016.)
|- ( ph -> A e. RR+ )   =>    |- ( ph -> ( log ` A ) e. RR )
Theorem	reeflogd 15610	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 15611	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 15612	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 15613	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 15614	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 15615	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 15616	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 15617	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 15618	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 15619	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 15620	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 15621	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 15622	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 15623	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 15624	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 15625	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 15626	Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. RR+ -> ( A ^c 1 ) = A )
Theorem	1cxp 15627	Value of the complex power function at one. (Contributed by Mario Carneiro, 2-Aug-2014.)
|- ( A e. CC -> ( 1 ^c A ) = 1 )
Theorem	ecxp 15628	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 15629	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 15630	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 15631	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 15632	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 15633	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 15634	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 15635	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 15636	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 15637	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 15638	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 15639	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	rpcxpmul2 15640	Product of exponents law for complex exponentiation. Variation on cxpmul 15639 with more general conditions on A and B when C is a nonnegative integer. (Contributed by Mario Carneiro, 9-Aug-2014.)
|- ( ( A e. RR+ /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )
Theorem	rpcxproot 15641	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 15642	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 15643	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 15644	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 15645	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 15646	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 15647	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 15648	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 15649	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 15650	Logarithm of a square root. (Contributed by Mario Carneiro, 5-May-2016.)
|- ( A e. RR+ -> ( log ` ( sqr ` A ) ) = ( ( log ` A ) / 2 ) )
Theorem	rpcxp0d 15651	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 15652	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 15653	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 15654	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 15655	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 15656	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 15657	Square root theorem over the complex numbers for the complex power function. Compare with resqrtth 11593. (Contributed by AV, 23-Dec-2022.)
|- ( A e. RR+ -> ( ( sqr ` A ) ^c 2 ) = A )
Theorem	cxprecd 15658	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	rpcxpmul2d 15659	Product of exponents law for complex exponentiation. Variation on cxpmul 15639 with more general conditions on A and B when C is a nonnegative integer. (Contributed by Mario Carneiro, 30-May-2016.)
|- ( ph -> A e. RR+ )   &    |- ( ph -> B e. CC )   &    |- ( ph -> C e. NN0 )   =>    |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )
Theorem	rpcxpcld 15660	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 15661	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 15662	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 15663	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 15664	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 15665	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 15666	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 15667	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 15668	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 ) ) )
Theorem	ltexp2d 15669	Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.)
|- ( ph -> A e. RR )   &    |- ( ph -> M e. ZZ )   &    |- ( ph -> N e. ZZ )   &    |- ( ph -> 1 < A )   =>    |- ( ph -> ( M < N <-> ( A ^ M ) < ( A ^ N ) ) )
11.2.4  Logarithms to an arbitrary base Define "log using an arbitrary base" function and then prove some of its properties. As with df-relog 15585 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 15670	Extend class notation to include the logarithm generalized to an arbitrary base.
class logb
Definition	df-logb 15671*	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 15672	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 15673	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 15674	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 15675	The logarithm of 1 to an arbitrary base B is 0. Property 1(b) of [Cohen4] p. 361. See log1 15593. (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 15676	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 15677	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 15678	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 15679	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 15680	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 15681	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 15682	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 15683	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 15684	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 15685	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 15686	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 15687	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 15688	The general logarithm function is monotone/increasing. See logleb 15602. (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 15689	The general logarithm function is strictly monotone/increasing. Property 2 of [Cohen4] p. 377. See logltb 15601. (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 15690	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 15691	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 15692	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 15693	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 15694	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 15695	Lemma for logbgcd1irrap 15697. 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 15696	Lemma for logbgcd1irrap 15697. 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 15697	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 15698	Example for logbgcd1irr 15694. The logarithm of nine to base two is not rational. Also see 2logb9irrap 15704 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 15699	The logarithm of a prime to a different prime base is not rational. For example, ( 2 logb 3 ) e. ( RR \ QQ ) (see 2logb3irr 15700). (Contributed by AV, 31-Dec-2022.)
|- ( ( X e. Prime /\ B e. Prime /\ X =/= B ) -> ( B logb X ) e. ( RR \ QQ ) )
Theorem	2logb3irr 15700	Example for logbprmirr 15699. The logarithm of three to base two is not rational. (Contributed by AV, 31-Dec-2022.)
|- ( 2 logb 3 ) e. ( RR \ QQ )