Is a linear equation always a straight line? I am studying Linear Algebra, and it just blows my mind how so many things in real life can be related to linear equations. I know this might be a stupid question, but is it correct to assume that all linear equations are just straight lines (with or with out slopes?), that none of the equations will ever have exponents, logarithms, etc.??
in this link, it uses an example with a variable being squared. how is that a linear equation?
I understand that you can have planes in 3 Dimensional spaces, but I guess my main question is, will there ever be a curve in a linear equation? 
 A: If your vector space is $\mathbb{R}^2$, then a linear equation in the variables $x$ and $y$ is effectively a line.
On the contrary, when you pass in $\mathbb{R}^3$, an equation of the form $ax+by+cz+d=0$ represents a plane. In order to obtain a straight line, you must consider two linear equations $ax+by+cz+d=0$ and $a'x+b'y+c'z+d'=0$ so that $(a,b,c)$ and $(a',b',c')$ are non proportional.
A: The zero locus of linearly related variables (by $f(x,y)=ax+by$ in the below) in $\mathbb{R}^2$ is a line through the origin. Meaning, the set of all $x$ and $y$ with 
$$
f(x,y)=ax+by=0
$$
This is a fancy way of saying that the graph of a linear map from $R^1$ to $R^1$ is a line through the origin. 
A: From Wiki:

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

This does not mean that we can't use nonlinear tools and concepts to help study these objects.  If that's too many negatives in one sentence, another way to say it is it's perfectly fine to use nonlinear tools to study linear objects.
As for your questions...

but is it correct to assume that all linear equations are just straight lines (with or with out slopes?)

Yes.  But once you go higher than 3 dimensions it becomes sort of... difficult to visualize.  For example, $2x_1 + 3x_2 - x_3 + 7x_4 = 0$ technically represents a line in 4 dimensions.  The concept of 2-dimensional slope is more or less generalized to higher dimensions with the gradient.

that none of the equations will ever have exponents, logarithms, etc.??

No.  For example, $\pi^5 x + (\ln 2) y = 0$ is a linear equation containing exponents and logs.  But $\pi x^5 + 2\ln y = 0$ is not a linear equation, which may be what you were getting at.

in this link, it uses an example with a variable being squared. how is that a linear equation?

I'm assuming you're talking about $y = \alpha x^2 + \beta x + \gamma$ in this answer to the question you linked to.  That's not a linear equation.  The answer author even said it's a second-order polynomial function of $x$.  This is an example of what I originally mentioned, i.e., using nonlinear tools to study linear objects.  In this case the linear object being studied is what the answer author goes on to talk about, which is a system of 100 equations in 3 unknowns.  The unknowns are the constants $\alpha, \beta, \gamma$.  And yes, that system does include $x_i^2$ in the matrix $X$, but the equations in the system are still linear because the variables in the system are $\alpha, \beta, \gamma$, not $x$.  It's the coefficients in the system that have the exponents, which is fine.

I understand that you can have planes in 3 Dimensional spaces, but I guess my main question is, will there ever be a curve in a linear equation? 

No, not in Euclidean geometry.  (I think lines are technically curved in some types of non-Euclidean geometry but that may be getting too off topic.)
