how to make maths equations start at the left?

documentclass{article}
usepackage{amsmath}
usepackage{showframe}   %% just for demo
begin{document}
  begin{flalign}
     lim_{ x to 0} frac{log {10 + log(x+ frac{1}{10})}}{x} = 10 && 
  end{flalign}
end{document}

If you want all your equations to start from left, add [fleqn] option to the documentclass. Further, you may want to set mathindent to zero.

documentclass[fleqn]{article}
usepackage{amsmath}
usepackage{showframe}   %% just for demo
setlengthmathindent{0pt}
begin{document}
  [
    lim_{ x to 0} frac{log {10 + log(x+ frac{1}{10})}}{x} = 10 
  ]
end{document}

Another method that works with all display math environments, if you don't want all equation to be aligned on the left margin, is the following.

We define a new command

newcommand{mathleft}{@fleqntrue@mathmargin0pt}

that simulates the option fleqn with mathindent=0pt, and a new command to restore the normal behavior

newcommand{mathcenter}{@fleqnfalse}

In the following MWE you can see how to use those commands:

documentclass{article}
usepackage{showframe} % just for the example
usepackage{amsmath}

makeatletter
newcommand{mathleft}{@fleqntrue@mathmargin0pt}
newcommand{mathcenter}{@fleqnfalse}
makeatother

begin{document}

mathleft
begin{equation}
lim_{ x to 0} frac{log {10 + log(x+ frac{1}{10})}}{x} = 10
end{equation}
mathcenter

begin{equation}
lim_{ x to 0} frac{log {10 + log(x+ frac{1}{10})}}{x} = 10
end{equation}

end{document} 

Output

Simplest way one can do for the fewer lines of equation is to put "&" which ever location you want to align from.

For Example,

begin{aligned}
1. &: E(a) = a 
2. &: E(aX) = aE(X) = a mu 
3. &: E(aX + b) = aE(X) + b 
4. &: E(X pm Y) = E(X) pm E(Y) 
5. &: E(aX pm bY) = aE(X) pm bE(Y) 
6. &: E(X^2) = sum x_i^2 P(x_i^2) = sum x_i^2 P(x_i) 
7. &: E(h(x)) = sum h(x_i) P(x_i) 
&:  where  h(x)  is  a   function 
8. &: E(X^3) = sum x_i^3 P(x_i^3) = sum x_i^3 P(x_i)
end{aligned}

Thanks.

There's also the nccmath package, which defines the fleqn environment for that. %oreover, this environment can take a optional argument to set the value of displayindent (0pt by default). Demo:

documentclass{article}
usepackage{xcolor}
usepackage{amsmath}
usepackage{nccmath}
usepackage{showframe} %% just for demo
renewcommand{ShowFrameColor}{color{red}}
renewcommand{ShowFrameLinethickness}{0.3pt}

begin{document}

Some text. Some more text. Some more text. Some more text. Still some more text.
  begin{fleqn}
     [ lim_{ x to 0} frac{log {10 + log(x+ frac{1}{10})}}{x} = 10 ]
  end{fleqn}

  begin{fleqn}[parindent]
  begin{equation}
  lim_{ x to 0} frac{log {10 + log(x+ frac{1}{10})}}{x} = 10
  end{equation}
  end{fleqn}

end{document}