Elementary Functions - 4.8 Identities

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.8.E1 Ln ⁡ ( z 1 ⁢ z 2 ) = Ln ⁡ z 1 + Ln ⁡ z 2 multivalued-natural-logarithm subscript 𝑧 1 subscript 𝑧 2 multivalued-natural-logarithm subscript 𝑧 1 multivalued-natural-logarithm subscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{Ln}\left(z_{1}z_{2}\right)=% \operatorname{Ln}z_{1}+\operatorname{Ln}z_{2}}}
\Ln@{z_{1}z_{2}} = \Ln@@{z_{1}}+\Ln@@{z_{2}}		 

ln(z[1]*z[2]) = ln(z[1])+ ln(z[2])

Log[Subscript[z, 1]*Subscript[z, 2]] == Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]
Failure Failure
Failed [25 / 100]
Result: 0.-6.283185308*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -1.5}


Result: 0.-6.283185308*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -.5}


Result: -.1e-9-6.283185308*I
Test Values: {z[1] = 1/2*3^(1/2)+1/2*I, z[2] = -2}


Result: .133199999e-10-6.283185307*I
Test Values: {z[1] = -1/2+1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [25 / 100]
Result: Complex[0.0, -6.283185307179587]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -1.5]}


Result: Complex[0.0, -6.283185307179587]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, 2], -0.5]}

... skip entries to safe data
4.8.E2 ln ⁡ ( z 1 ⁢ z 2 ) = ln ⁡ z 1 + ln ⁡ z 2 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\ln\left(z_{1}z_{2}\right)=\ln z_{1}+\ln z_{2}}}
\ln@{z_{1}z_{2}} = \ln@@{z_{1}}+\ln@@{z_{2}}		 - π ≤ ph ⁡ z 1 + ph ⁡ z 2 , ph ⁡ z 1 + ph ⁡ z 2 ≤ π formulae-sequence 𝜋 phase subscript 𝑧 1 phase subscript 𝑧 2 phase subscript 𝑧 1 phase subscript 𝑧 2 𝜋 {\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z_{1}+\operatorname{ph}z_% {2},\operatorname{ph}z_{1}+\operatorname{ph}z_{2}\leq\pi}} 
ln(z[1]*z[2]) = ln(z[1])+ ln(z[2])

Log[Subscript[z, 1]*Subscript[z, 2]] == Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]
Failure Failure Successful [Tested: 59] Successful [Tested: 75]
4.8.E3 Ln ⁡ z 1 z 2 = Ln ⁡ z 1 - Ln ⁡ z 2 multivalued-natural-logarithm subscript 𝑧 1 subscript 𝑧 2 multivalued-natural-logarithm subscript 𝑧 1 multivalued-natural-logarithm subscript 𝑧 2 {\displaystyle{\displaystyle\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname% {Ln}z_{1}-\operatorname{Ln}z_{2}}}
\Ln@@{\frac{z_{1}}{z_{2}}} = \Ln@@{z_{1}}-\Ln@@{z_{2}}		 

ln((z[1])/(z[2])) = ln(z[1])- ln(z[2])

Log[Divide[Subscript[z, 1],Subscript[z, 2]]] == Log[Subscript[z, 1]]- Log[Subscript[z, 2]]
Failure Failure
Failed [25 / 100]
Result: 0.-6.283185307*I
Test Values: {z[1] = -1/2+1/2*I*3^(1/2), z[2] = -1/2*3^(1/2)-1/2*I}


Result: 0.+6.283185307*I
Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)}


Result: .1e-9+6.283185307*I
Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1.5}


Result: -.1e-9+6.283185307*I
Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -.5}

... skip entries to safe data
Failed [25 / 100]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}


Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.8.E4 ln ⁡ z 1 z 2 = ln ⁡ z 1 - ln ⁡ z 2 subscript 𝑧 1 subscript 𝑧 2 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle\ln\frac{z_{1}}{z_{2}}=\ln z_{1}-\ln z_{2}}}
\ln@@{\frac{z_{1}}{z_{2}}} = \ln@@{z_{1}}-\ln@@{z_{2}}		 - π ≤ ph ⁡ z 1 - ph ⁡ z 2 , ph ⁡ z 1 - ph ⁡ z 2 ≤ π formulae-sequence 𝜋 phase subscript 𝑧 1 phase subscript 𝑧 2 phase subscript 𝑧 1 phase subscript 𝑧 2 𝜋 {\displaystyle{\displaystyle-\pi\leq\operatorname{ph}z_{1}-\operatorname{ph}z_% {2},\operatorname{ph}z_{1}-\operatorname{ph}z_{2}\leq\pi}} 
ln((z[1])/(z[2])) = ln(z[1])- ln(z[2])

Log[Divide[Subscript[z, 1],Subscript[z, 2]]] == Log[Subscript[z, 1]]- Log[Subscript[z, 2]]
Failure Failure
Failed [3 / 70]
Result: 0.+6.283185307*I
Test Values: {z[1] = 1/2-1/2*I*3^(1/2), z[2] = -1/2+1/2*I*3^(1/2)}


Result: 0.+6.283185308*I
Test Values: {z[1] = -1/2*3^(1/2)-1/2*I, z[2] = 1/2*3^(1/2)+1/2*I}


Result: 6.283185308*I
Test Values: {z[1] = 2, z[2] = -2}

Failed [11 / 86]
Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[Subscript[z, 1], Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]], Rule[Subscript[z, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.8.E5 Ln ⁡ ( z n ) = n ⁢ Ln ⁡ z multivalued-natural-logarithm superscript 𝑧 𝑛 𝑛 multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\operatorname{Ln}\left(z^{n}\right)=n\operatorname% {Ln}z}}
\Ln@{z^{n}} = n\Ln@@{z}		 

ln((z)^(n)) = n*ln(z)

Log[(z)^(n)] == n*Log[z]
Failure Failure
Failed [5 / 21]
Result: .133199999e-10-6.283185307*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 2, n = 3}


Result: .4399599996e-9-6.283185306*I
Test Values: {z = -1/2+1/2*I*3^(1/2), n = 3, n = 3}


Result: .4399599996e-9+6.283185306*I
Test Values: {z = 1/2-1/2*I*3^(1/2), n = 3, n = 3}


Result: .133199999e-10+6.283185307*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, n = 2, n = 3}

... skip entries to safe data
Failed [3 / 7]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.8.E6 ln ⁡ ( z n ) = n ⁢ ln ⁡ z superscript 𝑧 𝑛 𝑛 𝑧 {\displaystyle{\displaystyle\ln\left(z^{n}\right)=n\ln z}}
\ln@{z^{n}} = n\ln@@{z}		 - π ≤ n ⁢ ph ⁡ z , n ⁢ ph ⁡ z ≤ π formulae-sequence 𝜋 𝑛 phase 𝑧 𝑛 phase 𝑧 𝜋 {\displaystyle{\displaystyle-\pi\leq n\operatorname{ph}z,n\operatorname{ph}z% \leq\pi}} 
ln((z)^(n)) = n*ln(z)

Log[(z)^(n)] == n*Log[z]
Failure Failure
Failed [1 / 17]
Result: .4399599996e-9+6.283185306*I
Test Values: {z = 1/2-1/2*I*3^(1/2), n = 3, n = 3}

Failed [3 / 7]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.0, 6.283185307179586]
Test Values: {Rule[n, 3], Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.8.E7 ln ⁡ 1 z = - ln ⁡ z 1 𝑧 𝑧 {\displaystyle{\displaystyle\ln\frac{1}{z}=-\ln z}}
\ln@@{\frac{1}{z}} = -\ln@@{z}		 | ph ⁡ z | ≤ π phase 𝑧 𝜋 {\displaystyle{\displaystyle|\operatorname{ph}z|\leq\pi}} 
ln((1)/(z)) = - ln(z)

Log[Divide[1,z]] == - Log[z]
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.8.E8 Ln ⁡ ( exp ⁡ z ) = z + 2 ⁢ k ⁢ π ⁢ i multivalued-natural-logarithm 𝑧 𝑧 2 𝑘 𝜋 imaginary-unit {\displaystyle{\displaystyle\operatorname{Ln}\left(\exp z\right)=z+2k\pi% \mathrm{i}}}
\Ln@{\exp@@{z}} = z+2k\pi\iunit		 

ln(exp(z)) = z + 2*k*Pi*I

Log[Exp[z]] == z + 2*k*Pi*I
Failure Failure
Failed [21 / 21]
Result: -.1e-9-6.283185308*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}


Result: -.1e-9-12.56637062*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}


Result: -.1e-9-18.84955592*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k = 3, k = 3}


Result: 0.-6.283185308*I
Test Values: {z = -1/2+1/2*I*3^(1/2), k = 1, k = 3}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.0, -18.84955592153876]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, -18.84955592153876]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.8.E9 ln ⁡ ( exp ⁡ z ) = z 𝑧 𝑧 {\displaystyle{\displaystyle\ln\left(\exp z\right)=z}}
\ln@{\exp@@{z}} = z		 - π ≤ ℑ ⁡ z , ℑ ⁡ z ≤ π formulae-sequence 𝜋 𝑧 𝑧 𝜋 {\displaystyle{\displaystyle-\pi\leq\Im z,\Im z\leq\pi}} 
ln(exp(z)) = z

Log[Exp[z]] == z
Failure Failure Successful [Tested: 7] Successful [Tested: 7]
4.8.E10 exp ⁡ ( ln ⁡ z ) = exp ⁡ ( Ln ⁡ z ) 𝑧 multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\exp\left(\ln z\right)=\exp\left(\operatorname{Ln}% z\right)}}
\exp@{\ln@@{z}} = \exp@{\Ln@@{z}}		 

exp(ln(z)) = exp(ln(z))

Exp[Log[z]] == Exp[Log[z]]
Successful Successful - Successful [Tested: 7]
4.8.E10 exp ⁡ ( Ln ⁡ z ) = z multivalued-natural-logarithm 𝑧 𝑧 {\displaystyle{\displaystyle\exp\left(\operatorname{Ln}z\right)=z}}
\exp@{\Ln@@{z}} = z		 

exp(ln(z)) = z

Exp[Log[z]] == z
Successful Successful - Successful [Tested: 7]
4.8.E11 Ln ⁡ ( a z ) = z ⁢ Ln ⁡ a + 2 ⁢ k ⁢ π ⁢ i multivalued-natural-logarithm superscript 𝑎 𝑧 𝑧 multivalued-natural-logarithm 𝑎 2 𝑘 𝜋 imaginary-unit {\displaystyle{\displaystyle\operatorname{Ln}\left(a^{z}\right)=z\operatorname% {Ln}a+2k\pi\mathrm{i}}}
\Ln@{a^{z}} = z\Ln@@{a}+2k\pi\iunit		 

ln((a)^(z)) = z*ln(a)+ 2*k*Pi*I

Log[(a)^(z)] == z*Log[a]+ 2*k*Pi*I
Failure Failure
Failed [126 / 126]
Result: 0.-6.283185308*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 1, k = 3}


Result: 0.-12.56637062*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 2, k = 3}


Result: 0.-18.84955592*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 3, k = 3}


Result: 0.-6.283185308*I
Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2), k = 1, k = 3}

... skip entries to safe data
Failed [42 / 42]
Result: Complex[0.0, -18.84955592153876]
Test Values: {Rule[a, -1.5], Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, -18.84955592153876]
Test Values: {Rule[a, -1.5], Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.8.E12 ln ⁡ ( a z ) = z ⁢ ln ⁡ a + 2 ⁢ k ⁢ π ⁢ i superscript 𝑎 𝑧 𝑧 𝑎 2 𝑘 𝜋 imaginary-unit {\displaystyle{\displaystyle\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i}}}
\ln@{a^{z}} = z\ln@@{a}+2k\pi\iunit		 

ln((a)^(z)) = z*ln(a)+ 2*k*Pi*I

Log[(a)^(z)] == z*Log[a]+ 2*k*Pi*I
Failure Failure
Failed [126 / 126]
Result: 0.-6.283185308*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 1}


Result: 0.-12.56637062*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 2}


Result: 0.-18.84955592*I
Test Values: {a = -1.5, z = 1/2*3^(1/2)+1/2*I, k = 3}


Result: 0.-6.283185308*I
Test Values: {a = -1.5, z = -1/2+1/2*I*3^(1/2), k = 1}

... skip entries to safe data
Failed [126 / 126]
Result: Complex[0.0, -6.283185307179586]
Test Values: {Rule[a, -1.5], Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, -12.566370614359172]
Test Values: {Rule[a, -1.5], Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
4.8.E13 ln ⁡ ( a x ) = x ⁢ ln ⁡ a superscript 𝑎 𝑥 𝑥 𝑎 {\displaystyle{\displaystyle\ln\left(a^{x}\right)=x\ln a}}
\ln@{a^{x}} = x\ln@@{a}		 a > 0 𝑎 0 {\displaystyle{\displaystyle a>0}} 
ln((a)^(x)) = x*ln(a)

Log[(a)^(x)] == x*Log[a]
Successful Failure - Successful [Tested: 9]
4.8.E14 a z 1 ⁢ a z 2 = a z 1 + z 2 superscript 𝑎 subscript 𝑧 1 superscript 𝑎 subscript 𝑧 2 superscript 𝑎 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle a^{z_{1}}a^{z_{2}}=a^{z_{1}+z_{2}}}}
a^{z_{1}}a^{z_{2}} = a^{z_{1}+z_{2}}		 a ≠ 0 𝑎 0 {\displaystyle{\displaystyle a\neq 0}} 
(a)^(z[1])* (a)^(z[2]) = (a)^(z[1]+ z[2])
 
(a)^(Subscript[z, 1])* (a)^(Subscript[z, 2]) == (a)^(Subscript[z, 1]+ Subscript[z, 2])
 Skipped - no semantic math Skipped - no semantic math - -
4.8.E15 a z ⁢ b z = ( a ⁢ b ) z superscript 𝑎 𝑧 superscript 𝑏 𝑧 superscript 𝑎 𝑏 𝑧 {\displaystyle{\displaystyle a^{z}b^{z}=(ab)^{z}}}
a^{z}b^{z} = (ab)^{z}		 - π ≤ ph ⁡ a + ph ⁡ b , ph ⁡ a + ph ⁡ b ≤ π formulae-sequence 𝜋 phase 𝑎 phase 𝑏 phase 𝑎 phase 𝑏 𝜋 {\displaystyle{\displaystyle-\pi\leq\operatorname{ph}a+\operatorname{ph}b,% \operatorname{ph}a+\operatorname{ph}b\leq\pi}} 
(a)^(z)* (b)^(z) = (a*b)^(z)
 
(a)^(z)* (b)^(z) == (a*b)^(z)
 Skipped - no semantic math Skipped - no semantic math - -
4.8.E16 e z 1 ⁢ e z 2 = e z 1 + z 2 superscript 𝑒 subscript 𝑧 1 superscript 𝑒 subscript 𝑧 2 superscript 𝑒 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle e^{z_{1}}e^{z_{2}}=e^{z_{1}+z_{2}}}}
e^{z_{1}}e^{z_{2}} = e^{z_{1}+z_{2}}		 

exp(z[1])*exp(z[2]) = exp(z[1]+ z[2])
 
Exp[Subscript[z, 1]]*Exp[Subscript[z, 2]] == Exp[Subscript[z, 1]+ Subscript[z, 2]]
 Skipped - no semantic math Skipped - no semantic math - -
4.8.E17 ( e z 1 ) z 2 = e z 1 ⁢ z 2 superscript superscript 𝑒 subscript 𝑧 1 subscript 𝑧 2 superscript 𝑒 subscript 𝑧 1 subscript 𝑧 2 {\displaystyle{\displaystyle(e^{z_{1}})^{z_{2}}=e^{z_{1}z_{2}}}}
(e^{z_{1}})^{z_{2}} = e^{z_{1}z_{2}}		 - π ≤ ℑ ⁡ z 1 , ℑ ⁡ z 1 ≤ π formulae-sequence 𝜋 subscript 𝑧 1 subscript 𝑧 1 𝜋 {\displaystyle{\displaystyle-\pi\leq\Im z_{1},\Im z_{1}\leq\pi}} 
(exp(z[1]))^(z[2]) = exp(z[1]*z[2])
 
(Exp[Subscript[z, 1]])^(Subscript[z, 2]) == Exp[Subscript[z, 1]*Subscript[z, 2]]
 Skipped - no semantic math Skipped - no semantic math - -
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