solve logarithmic equation without numerical methods

Question

Is there algebraic method to solve following equation for $x$: $$ a x + b \ln x + c = 0 $$ with $a , b , c$ constants without using numerical methods and ln means natural logarithm.

Answer 1

$$ ax+b\ln(x)+c=0\\ ax+b\ln\left(\frac abx\right)+c-b\ln\left(\frac ab\right)=0\\ \frac abx+\ln\left(\frac abx\right)+\frac cb-\ln\left(\frac ab\right)=0\\ {(\frac{a}{b})x}\,e^{{(\frac{a}{b})x}}=(\frac{a}{b})\,e^{-\frac{c}{b}} $$ The Lambert-W function is the inverse of $xe^x$. $\Rightarrow$ $$ \\{(\frac{a}{b})x}=\mathrm{W}\left((\frac{a}{b})e^{\frac{-c}{b}}\right) $$ $\Rightarrow$ $$ x=(\frac{b}{a})\mathrm{W}\left((\frac{a}{b})e^{\frac{-c}{b}}\right) $$