 Another type of function we could look at are log and exponential functions. So let's consider the graph of the function f of x equals 2 to power x. So we'll graph this function by finding the x and y intercepts, and possibly by finding additional asymptotes. So the y-intercept occurs when x is equal to zero. So our graph is y equals f of x, and equals means replaceable. Since f of x is 2 to power x, we'll replace. Since equals means replaceable, x equals zero, we'll replace. And solve for y. And remember the y-intercept is a point, so we have to specify both the x and y coordinates. And so on the graph of y equals f of x, we have the y-intercept at zero, one. So we can also find the x-intercept. The x-intercept occurs when y is equal to zero. So we have y equals f of x, and we'll replace. Now we'll try to solve this equation. We can hit both sides with a log. But remember the log of c is only defined for c greater than zero. So log of zero is undefined, and since log of zero is undefined, there is no solution and no x-intercept. What about n-behavior? Well let's see what happens to 2 to power x as x gets large. So we can evaluate 2 to the tenth. How about 2 to the one-hundredth power? And for good measure, how about 2 to the power one-thousand? And so we see that as x goes to infinity, 2 to the power x also goes to infinity. How about in the other direction? If x equals negative twenty, then 2 to the negative twenty is. Or how about 2 to the power negative one-fifty-seven? And if we take even more negative powers of x, we see that 2 to the power x goes to zero. And so as x goes to minus infinity, 2 to the power x goes to zero. Remember if it's not written down, it didn't happen. As x goes to infinity, 2 to the x goes to infinity. But 2 to the x is f of x, and f of x is y, and so this means that as x goes to infinity, y goes to infinity. Likewise, as x goes to minus infinity, 2 to the x goes to zero. But 2 to the x is f of x, f of x is y, and so this means that y goes to zero. And this means that y equals zero is a horizontal asymptote. So we'll sketch a graph that has y intercepted zero, one. No x intercept, which means we never cross the x-axis. As x goes to infinity, our y values get larger as well. And as x goes to minus infinity, our y values go to zero. Now we can get all exponential functions as long as we remember the rules of exponents. And the two important rules here are a to the power m to the power n is a to the power mn, and a to the power minus m is the same as 1 over a to the power m. And if we have these rules, we can use a graph transformation to produce the graph of any exponential function from the graph of y equals 2 to the power x. For example, if I want to sketch the graph of y equals 1 half to power x, I can do that as follows. What we want is to write 1 half as 2 to some power. So we write down 1 half equals 2, and that's wrong. But that's OK, we can fix it. So our exponent must be negative 1. Equals means replaceable. So our graph y equals 1 half to the power x, we can replace 1 half with 2 to the power minus 1. We can apply our rules of exponents to rewrite this as 2 to the power minus x. Now the transformation that would replace x with minus x is going to be a reflection across the y-axis. And so the graph of y equals 1 half to the x can be produced by reflecting the graph of y equals 2 to the x across the y-axis. So we'll draw our graph of y equals 2 to the x, then reflect it across the y-axis. And don't forget to label the graph. Or how about the graph of y equals 5 to the x? Again, we want to write 5 as 2 to some power, and we can do that. We can solve. And that power's got to be around 2.322. Equals means replaceable. So in y equals 5 to the x, 5 is 2 to the n, n is 2.322. So that says y is 2 to the power 2.322 raised to the x. Our rules of exponents say we can multiply the two exponents. And so this tells us that the graph of y equals 5 to the x can be produced from the graph of y equals 2 to the x by compressing it horizontally by a factor of 2.322. So we'll take our graph of y equals 2 to the x and squeeze it in a little bit. And if it's not written down, it didn't happen, so we need to label this new graph. What about our graph of g of x equals log to base 2 of x? Now remember that log and exponential functions are inverses of each other. So if we reflect y equals 2x across the line y equals x, then any point x, y in the original graph will become the point capital X, capital Y on the reflected graph where capital X is our y value and capital Y is our x value. So the equation y equals 2 to the x will replace. Now remember that our definition of logarithms tells us that if we have a power equation, we can immediately rewrite it as a log equation. So we can rewrite our power equation as a log equation. And so this reflection across the line y equals x will give us the graph of y equals log 2 of x. Now while it's easy enough to do this reflection geometrically, it's worth remembering what we're actually doing algebraically as well. Our transformation has essentially switched the x and y coordinates. So everything we know about the exponential function becomes something we know for the logarithmic function if we swap the x and y values. So on the graph of y equals f of x we have the y-intercept at 0, 1. Switching things around gives us the x-intercept at 1, 0. We said there was no x-intercept, so we'll replace there is no y-intercept. We said that as x goes to infinity, y goes to infinity. So replacing x and y with y and x as y goes to infinity, x goes to infinity. Now we typically like to express things in terms of x, so we could reword this and just say that as x goes to infinity, y goes to infinity. On our exponential function, as x goes to minus infinity, y goes to 0. So we'll replace as y goes to minus infinity, x goes to 0. And again we like to lead with the x, so we'll take this statement and reword it as x goes to 0, y goes to minus infinity. Y equals 0 is a horizontal asymptote, so now x equals 0 is a horizontal asymptote. Wait, x equals 0 isn't horizontal, that's a vertical asymptote. And so now we know what the graph of y equals log 2 of x is, and by extension every logarithmic graph is going to look like some variation of this.