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I'm trying to find the shortest distance between a point and a logarithmic line. So far I've applied the distance formula, found the derivative of the distance equation and made it equal to zero. That's how I got the equation in the title: $$2x^2 - 5x + 64log(x) + 776 = 0.$$ From this point I haven't been able to figure out how to solve for $x$. So the question is, how to I solve for $x$ in this equation?

I am pretty sure that it is impossible to solve for the equation algebraically. Let $f(x) = 2x^2 - 5x +64log x+776$. We are trying to find when $f(x) = 0$. Note that the domain of $f$ is $(0,infty)$ because of the logarithm( which I assume is the natural logarithm i.e. to the base $e$).

Going for it analytically, first we note that the derivative is $f' = 4x + frac64x - 5 = frac4x^2 - 5x+64x$ which is strictly positive on $(0,infty)$ since the numerator has discriminant negative , hence is always positive, while the denominator is always positive.

Also, from the fact that $f(x) to -infty$ as $x to 0^+$ and $f(1) >0$, we see that any root is between $0$ and $1$.

We can actually do better : note that if $x = e^y$, where $y < 0$ since we now want $0<x<1$, then we have : $$2x^2 - 5x = 2(x^2 - 2.5x) = 2((x-1.25)^2) - frac258$$

Therefore, we see that on $[0,1]$ that this part is between $0$ and $-3$.

So if equality holds, then in particular, $64y + 776$ is between $0$ and $3$, so we get $frac-77664 leq y leq frac-77364$, which after exponentiation gives $ 5.422leq 10^6x leq 5.682$. At least we get the number of significant zeros and the first significant non-zero digit, using at most two exponential operations.

first the derivation of this equation is greater than zero

$y=2x^2-5x+64log(x)+776$ so $fracdydx=4x-5+frac64x=frac4x^2-5x+64x$ in $4x^2-5x+64geq 0$ since $Delta leq 0$ so always $fracdydx>0$

also $y(0^+)=-infty$ and there are some $ygeq 0$ so you have a one root and no more. (since $y$ is increasing and exist $a$ and $b$ such that $f(a)f(b)<0$و Intermediate value theorem)

R code

 y<-function(x)
 2*x^2-5*x+64*log(x)+776
 
 uniroot(y,c(5.414414e-06,5.423423e-06))
 $root
 [1] 5.423423e-06

 $f.root
 [1] 0.01383579

 $iter
 [1] 0

 $init.it
 [1] NA

 $estim.prec
 [1] 9.009e-09

so you sure have a unique answer