Johann Heinrich Lambert, born in 1728 in Mulhouse, left school at 12 and worked various jobs while continuing his studies. He traveled Europe as a tutor, meeting many mathematicians. In 1763, he joined the Prussian Academy of Sciences in Berlin, supported by Frederick II of Prussia and befriended Euler. Lambert worked prolifically until his death in 1777.

For more information, you can visit Johann Heinrich Lambert - Wikipedia

• To solve the equation

  $$ x^2 =k,\ k\in \mathbb{R} $$

  we simply apply the square root to both sides, assuming $x>0$:

  $$ x =\sqrt{k},\ k\in \mathbb{R} $$

• For the equation

  $$ a^x=k,\ k>0 $$

  we use logarithms with base $a>0,\ a\not=1$:

  $$ x=\log_ak $$

However, solving the equation

$$ xe^x=k $$

is not straightforward. This is where Lambert’s $W$ function comes into play. It satisfies:

$$ z = W(ze^{z}),\ \forall z\in \mathbb{C} $$

This is the inverse function of

$$ f(w) = we^w,\ \forall w\in \mathbb{C} $$

Since $f$ is not injective, then $W$ is multivalued. For the real case $W$ is only defined for $x>-1/e$ and it is double-valued in $(-1/e,\ 0)$.

Let us see an example

$$ 2^x+x=5 $$

Our objective is to achieve something like $k=xe^x$:

$$ \begin{align*} 2^x+x=5&\iff 2^x=5-x\\ &\iff 1=({5-x})2^{-x}\\ &\iff 2^5=(5-x)2^{5-x}\\ &\iff 32 = v\cdot 2^v\\ &\iff 32\ln 2=(\ln 2\cdot v)e^{\ln2 v}\\ &\iff W(32\ln 2)=W((\ln 2\cdot v)e^{\ln2 v})=\ln2\cdot v\\ &\iff x=5-\frac{W(32\ln 2)}{\ln 2} \end{align*} $$

For more information and examples, you can visit Lambert W function - Wikipedia and Función W de Lambert - Wikipedia, la enciclopedia libre.