We want to graph the exponential function y equals two raised to the power of x
and the log function y equals log base two of x on the same coordinate plane,
and then we'll discuss the relationship between the two functions.
To graph these functions, we'll graph them by hand by completing a table of values.
So for y equals two raised to the power of x, notice how I've already selected x values.
Now we'll perform substitution to determine the corresponding y value.
So when x is equal to zero, y is equal to two raised to the power of zero.
Anything raised to the power of zero is equal to one, so y is equal to one.
When x is equal to one, y is equal to two to the first, or two. When x is equal to two,
y is equal to two to the second power, or two squared, which is equal to four.
When x is three, y is equal to two to the third, which is equal to eight.
And notice I also selected two negative values for x. When x is equal to negative one, we would
have y equals two raised to the power of negative one, which would be one over two to the first,
or just one half. When x is negative two, y is equal to two to the negative two,
which is equal to one over two squared, which would be one fourth.
Let's go ahead and plot these points on the coordinate plane and sketch our graph.
We'll sketch the exponential function in blue. So we have the point zero one,
which is the y-intercept. We have the point one two, the point two four, the point three eight,
negative one one half, which would be here, and negative two one fourth, which would be here.
So our function looks something like this.
Notice how we have exponential growth because the base is greater than one.
We have a horizontal asymptote along the x-axis, or y equals zero.
Notice the domain, or all possible values of x, would be all real numbers,
but the range, or the y values, would be y is always greater than zero
because the function is always above the x-axis.
Now, to sketch the log function, we're going to rewrite this in exponential form. Remember,
y equals log base two of x is equivalent to two raised to the power of y must equal x.
So to complete a table of values for a log function, it's easier to rewrite it as an
exponential function. And then instead of selecting x values, we select y values
and then determine the value of x. But before we do this, notice this equation here
and this equation here look very similar except the x and y variables have been interchanged.
So because y is now the exponent, we'll select y values and determine x.
Notice how these y values here are the same as the x values we had here for the exponential function.
So when y is equal to zero, we have two to the zero equals x, so x is one. When y is one,
two to the first equals x, which is two. When y is two, two squared equals x, which would be four.
When y is three, two to the third equals x, or eight equals x.
When y equals negative one, we'd have two to the negative one equals x, which is one half.
And when y is negative two, we'd have two to the negative two equals x, which is one fourth.
Now let's go ahead and sketch the log function in red.
So we have the point one zero, or the x-intercept, two one, four two, and eight three,
as well as one half negative one and one fourth negative two.
So our log function looks something like this.
Now, there are several things to notice about this function. The log function has a vertical
asymptote, which would be the y-axis, where the vertical line x equals zero.
Notice the domain of this function would be x greater than zero, and the range of this function
would actually be all reals because this function goes down forever as well as goes up forever.
Now, if we compare these two functions, notice how, looking at the table of values,
the x and y coordinates have been interchanged. As well as when both equations are written in
exponential form, notice how the exponential function has a horizontal asymptote of y
equals zero. The log function has a vertical asymptote of x equals zero,
and notice how the domain and range have also been interchanged between the two functions.
So we should recognize that these two functions are inverses of one another.
Another way to show that these two functions are inverses is to graph the line y equals x
and show these two functions are symmetrical across this line.
Well, this line has a y-intercept of zero and a slope of one,
so it would pass through these points here and look something like this.
So notice if we folded these two functions across this line,
they would match up perfectly, again showing they are inverses of one another.
Okay, I hope you found this video helpful.