How to solve equations with both logarithms and square roots, like this: $ax+blog(x)+csqrtx+d=0$

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I have an equation that looks like this:



$$ax+b log(x)+csqrt x+d=0$$



I know that an equation without the $sqrt x$ can be solved using the Lambert's W function (How to solve equations with logarithms, like this: $ ax + blog(x) + c=0$). I am having trouble solving the equation with $sqrt x$, is there a (quasi) closed-form solution for this?




logarithms radicals transcendental-equations




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  • 1




    I can't prove it but, no, there is no closed-form, even with Lambert.
    – Yves Daoust
    Jul 25 at 18:39










  • Do you have an example for the parameters $$a,b,c,d$$?
    – Dr. Sonnhard Graubner
    Jul 25 at 18:47










  • The exact equation is: $8x +r log (4x) +6 sqrt x + 1=0$ where $r$ is a parameter.
    – Hooman
    Jul 25 at 20:10















up vote
0
down vote

favorite












I have an equation that looks like this:



$$ax+b log(x)+csqrt x+d=0$$



I know that an equation without the $sqrt x$ can be solved using the Lambert's W function (How to solve equations with logarithms, like this: $ ax + blog(x) + c=0$). I am having trouble solving the equation with $sqrt x$, is there a (quasi) closed-form solution for this?




logarithms radicals transcendental-equations




share|cite|improve this question

















  • 1




    I can't prove it but, no, there is no closed-form, even with Lambert.
    – Yves Daoust
    Jul 25 at 18:39










  • Do you have an example for the parameters $$a,b,c,d$$?
    – Dr. Sonnhard Graubner
    Jul 25 at 18:47










  • The exact equation is: $8x +r log (4x) +6 sqrt x + 1=0$ where $r$ is a parameter.
    – Hooman
    Jul 25 at 20:10













up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have an equation that looks like this:



$$ax+b log(x)+csqrt x+d=0$$



I know that an equation without the $sqrt x$ can be solved using the Lambert's W function (How to solve equations with logarithms, like this: $ ax + blog(x) + c=0$). I am having trouble solving the equation with $sqrt x$, is there a (quasi) closed-form solution for this?




logarithms radicals transcendental-equations




share|cite|improve this question













I have an equation that looks like this:



$$ax+b log(x)+csqrt x+d=0$$



I know that an equation without the $sqrt x$ can be solved using the Lambert's W function (How to solve equations with logarithms, like this: $ ax + blog(x) + c=0$). I am having trouble solving the equation with $sqrt x$, is there a (quasi) closed-form solution for this?





logarithms radicals transcendental-equations





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 25 at 18:25









amWhy

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189k25219431









asked Jul 25 at 18:21









Hooman

11




11







  • 1




    I can't prove it but, no, there is no closed-form, even with Lambert.
    – Yves Daoust
    Jul 25 at 18:39










  • Do you have an example for the parameters $$a,b,c,d$$?
    – Dr. Sonnhard Graubner
    Jul 25 at 18:47










  • The exact equation is: $8x +r log (4x) +6 sqrt x + 1=0$ where $r$ is a parameter.
    – Hooman
    Jul 25 at 20:10













  • 1




    I can't prove it but, no, there is no closed-form, even with Lambert.
    – Yves Daoust
    Jul 25 at 18:39










  • Do you have an example for the parameters $$a,b,c,d$$?
    – Dr. Sonnhard Graubner
    Jul 25 at 18:47










  • The exact equation is: $8x +r log (4x) +6 sqrt x + 1=0$ where $r$ is a parameter.
    – Hooman
    Jul 25 at 20:10








1




1




I can't prove it but, no, there is no closed-form, even with Lambert.
– Yves Daoust
Jul 25 at 18:39




I can't prove it but, no, there is no closed-form, even with Lambert.
– Yves Daoust
Jul 25 at 18:39












Do you have an example for the parameters $$a,b,c,d$$?
– Dr. Sonnhard Graubner
Jul 25 at 18:47




Do you have an example for the parameters $$a,b,c,d$$?
– Dr. Sonnhard Graubner
Jul 25 at 18:47












The exact equation is: $8x +r log (4x) +6 sqrt x + 1=0$ where $r$ is a parameter.
– Hooman
Jul 25 at 20:10





The exact equation is: $8x +r log (4x) +6 sqrt x + 1=0$ where $r$ is a parameter.
– Hooman
Jul 25 at 20:10
















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