![]() | 1 | initial version |
Setting $Y:=-(\ln X + 0.0064) = -\ln(e^{0.0064}X)$, we get an equation: $$Ye^Y = -\frac{0.08}{e^{0.0064}},$$ implying that $Y = W( -\frac{0.08}{e^{0.0064}} )$, where $W$ is Lambert W function. Then $$X = e^{- W( -\frac{0.08}{e^{0.0064}} ) - 0.0064}.$$
In Sage:
exp(-lambert_w(-0.08/exp(0.0064)) - 0.0064 )
gives 1.08359901378819
. You can also try other branches of $W$, like:
exp(-lambert_w(-0.08/exp(0.0064),1) - 0.0064 )
which gives 0.563525163706476
, etc.
![]() | 2 | No.2 Revision |
Setting $Y:=-(\ln X + 0.0064) = -\ln(e^{0.0064}X)$, we get an equation:
$$Ye^Y = -\frac{0.08}{e^{0.0064}},$$
implying that $Y = W( -\frac{0.08}{e^{0.0064}} )$, where $W$ is Lambert W function. function. Then
$$X = e^{- W( -\frac{0.08}{e^{0.0064}} ) - 0.0064}.$$
In Sage:
exp(-lambert_w(-0.08/exp(0.0064)) - 0.0064 )
gives 1.08359901378819
. You can also try other branches of $W$, like:
exp(-lambert_w(-0.08/exp(0.0064),1) - 0.0064 )
which gives 0.563525163706476
, etc.
![]() | 3 | No.3 Revision |
Setting $Y:=-(\ln X + 0.0064) = -\ln(e^{0.0064}X)$, we get an equation: $$Ye^Y = -\frac{0.08}{e^{0.0064}},$$ implying that $Y = W( -\frac{0.08}{e^{0.0064}} )$, where $W$ is Lambert W function. Then $$X = e^{- W( -\frac{0.08}{e^{0.0064}} ) - 0.0064}.$$
In Sage:
exp(-lambert_w(-0.08/exp(0.0064)) - 0.0064 )
gives 1.08359901378819
. You can also try other branches of $W$, like:
exp(-lambert_w(-0.08/exp(0.0064),1) exp(-lambert_w(1, -0.08/exp(0.0064)) - 0.0064 )
which gives
, etc.0.56352516370647658.6383141226570 - 91.0209263484521*I
![]() | 4 | No.4 Revision |
Setting $Y:=-(\ln X + 0.0064) = -\ln(e^{0.0064}X)$, we get an equation: $$Ye^Y = -\frac{0.08}{e^{0.0064}},$$ implying that $Y = W( -\frac{0.08}{e^{0.0064}} )$, where $W$ is Lambert W function. Then $$X = e^{- W( -\frac{0.08}{e^{0.0064}} ) - 0.0064}.$$
In Sage:
exp(-lambert_w(-0.08/exp(0.0064)) - 0.0064 )
gives 1.08359901378819
. You can also try other branches of $W$, like:
exp(-lambert_w(1, exp(-lambert_w(-1, -0.08/exp(0.0064)) - 0.0064 )
which gives
, etc.58.6383141226570 - 91.0209263484521*I48.6340483659129
![]() | 5 | No.5 Revision |
Setting $Y:=-(\ln X + 0.0064) = -\ln(e^{0.0064}X)$, we get an equation: $$Ye^Y = -\frac{0.08}{e^{0.0064}},$$ implying that $Y = W( -\frac{0.08}{e^{0.0064}} )$, where $W$ is Lambert W function. Then $$X = e^{- W( -\frac{0.08}{e^{0.0064}} ) - 0.0064}.$$
In Sage:Sage we can define a function that computes $X$ depending on a branch of $W$:
exp(-lambert_w(-0.08/exp(0.0064)) - 0.0064 )
gives 1.08359901378819
. You can also try other branches of $W$, like:
exp(-lambert_w(-1, def X(branch=0):
exp(-lambert_w(branch, -0.08/exp(0.0064)) - 0.0064 )
which Then X(0)
gives 1.08359901378819
, X(-1)
gives 48.6340483659129
, etc.and other branches give various complex solutions.
![]() | 6 | No.6 Revision |
Setting $Y:=-(\ln X + 0.0064) = -\ln(e^{0.0064}X)$, we get an equation: $$Ye^Y = -\frac{0.08}{e^{0.0064}},$$ implying that $Y = W( -\frac{0.08}{e^{0.0064}} )$, where $W$ is Lambert W function. Then $$X = e^{- W( -\frac{0.08}{e^{0.0064}} ) - 0.0064}.$$
In Sage we can define a function that computes $X$ depending on a branch of $W$:
def X(branch=0):
return exp(-lambert_w(branch, -0.08/exp(0.0064)) - 0.0064 )
Then X(0)
gives 1.08359901378819
, X(-1)
gives 48.6340483659129
, and other branches give various complex solutions.