 As you saw with the graphs of logarithmic functions, a logarithm does not have a complete domain. What I mean is not all real numbers work inside of a logarithm. So, for example, if we have our function f of x, which is equal to adx, it's just a standard exponential function. We can see that the domain of f is going to be all real numbers, sure enough. But the range of f is going to only be zero to infinity. It's only going to be positive numbers. Now, when it comes to the inverse function, these things get swapped around. So, the domain of f actually becomes the range of f inverse. So, every number can get hit by a logarithm. Every y-coordinate could be part of a logarithm. But on the other hand, the domain of f inverse is going to be the range of the function. So, in this case, the logarithm, right, the domain of a logarithm here, logarithm base a, this is only going to be zero to infinity. So, that is, when you take something like log base a of x, we have to have that x is greater than zero, greater than zero. The quality is not allowed, negative is not allowed here. Much like taking the square root of a negative, if you take the logarithm of a negative, it actually produces, it's going to produce imaginary output, just for a fun little fact right here. If you take the log, let's say the natural log of negative one, this is actually equal to pi times i, for which I won't give any more explanation about it. But the point is, we can't take logarithms of negative numbers because it actually produces imaginary numbers. And also, if you take the log base a of zero, this really, if it was anything, it would have to be like negative infinity, which is not a number. So, we have to rule out anything other than positive operands with our logarithm. So, if, for example, we have a function f of x equals the natural, or the log base two of x plus three, very similar to finding domains of square root functions, we have to make sure that the operand, right, the input is positive. So, to find the domain here, we have to solve the inequality, x plus three is greater than zero, which is easy enough just to track three from both sides, we get x is greater than negative three, and so then the domain of our function f here would be negative three to infinity, where we do not include negative three here, because the log base three of zero is undefined in this situation. So, the domain would be negative three to infinity. The difficulty here of solving this comes down to really how complicated is the argument of the logarithm that is this function inside of it. Well, if we have f of x equals the common log of five minus two x, to find the domain, we have to solve the inequality five minus two x is greater than zero, for which then we can add two x to both sides, we get five is greater than two x, we could divide both sides by two, we get five halves is greater than x, or if we flip things around, we get x is less than five halves, in which case this suggests to us that the domain of f here is going to equal negative infinity up to five halves, not including five halves itself. And so the difficulty, you know, like I said before, it just comes out how complicated is the function inside of the logarithm. If we have f of x this time is equal to the natural log of four minus x squared, to find the domain, we have to then solve the inequality four minus x squared is greater than zero. This is a quadratic inequality. I think we're better trying to solve this by factoring four minus x squared is a difference of squares. It would factor as two minus x and two plus x. Notice this tells me that my markers are going to be x equals two and negative two. Okay. So if I were to graph this, that is, I want to graph the quadratic equation y equals four minus x squared. Well then I would get these two markers, you get negative two and positive two. Notice how the leading coefficient actually has a negative sign in front of it. So that means the parabola would concave downward. Oh boy, let's try that again. That's a little bit better. It doesn't have to be a really great picture though. But we want things that are greater than zero, right? So greater than zero here means we're looking for on the picture, those things which are above the x-axis. And so we're going to see that's going to happen in this sector right here because our function here is above the x-axis. So what that means for us is that the domain of our function f in this situation is going to be negative two to two. So we get all numbers between negative two and two. But the x-intercepts of that parabola are not included because that would be the natural log of zero which is undefined. And this shows you then how you can solve for the domain of any logarithmic function. It comes down to how difficult, you know, if you have your function or the natural log of some function g of x, finding the domain really comes down to how easy is it to solve the inequality g of x is greater than zero.