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Let $x=1+y\log(x)$. How do we solve this equation for $x$? I know that $x$ can be represented in terms of the product log function (or the Lambert's W function) of $y$. My question is, can we solve it without the help of this product log function?

Thank you for your time and help.

Mhenni Benghorbal
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nicole
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2 Answers2

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I actually want to know how do you write x in terms of y without the Lambert's W function.

The function $y(x)=\dfrac{x-1}{\ln x}$ possesses no elementary inverse.

Lucian
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You could use a series in powers of $y-1$:

$$ x = 1+2\, \left( y-1 \right) +{\frac {2}{3}} \left( y-1 \right) ^{2}-{ \frac {2}{9}} \left( y-1 \right) ^{3}+{\frac {14}{135}} \left( y-1 \right) ^{4}-{\frac {22}{405}} \left( y-1 \right) ^{5}+{\frac {82}{ 2835}} \left( y-1 \right) ^{6}-{\frac {86}{6075}} \left( y-1 \right) ^ {7}+{\frac {622}{127575}} \left( y-1 \right) ^{8}+{\frac {1438}{ 1148175}} \left( y-1 \right) ^{9}-{\frac {1025966}{189448875}} \left( y-1 \right) ^{10}+{\frac {32909314}{3978426375}} \left( y-1 \right) ^{ 11}-{\frac {1584376606}{155158628625}} \left( y-1 \right) ^{12}+{ \frac {1068478318}{93095177175}} \left( y-1 \right) ^{13}-{\frac { 3415445666}{279285531525}} \left( y-1 \right) ^{14}+{\frac { 87807997126}{6982138288125}} \left( y-1 \right) ^{15}-{\frac { 52325133235058}{4154372281434375}} \left( y-1 \right) ^{16}+{\frac { 153840284803882}{12463116844303125}} \left( y-1 \right) ^{17}-{\frac { 25287193065300454}{2131192980375834375}} \left( y-1 \right) ^{18}+\ldots $$ which appears to converge for $|y-1| < 1$.

Robert Israel
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