Question Video: Determining the Solution of an Exponential Function Mathematics

Video Transcript

Use the graphs below to answer the following question. True or False: The equation two to the power of 𝑥 equals 𝑥 has no solution.

Let’s begin by having a look at the two functions which we can see below. We might be familiar with this straight-line graph. It’s the equation 𝑦 equals 𝑥, or the function 𝑓 of 𝑥 equals 𝑥. In this function, every input value of 𝑥 is equal to the output value 𝑦. And so what is the other function that is also on this graph? The shape of this function indicates that it may be an exponential function. An exponential function is of the form 𝑓 of 𝑥 equals 𝑎 times 𝑏 to the power of 𝑥, where 𝑎 and 𝑏 are real constants, 𝑏 is positive, and 𝑏 is not equal to one. The shape of this unknown function in blue is one of exponential growth. And that means that we can say that the value of 𝑏 must be greater than one. In fact, this graph will be that of 𝑦 equals two to the power of 𝑥.

We can check this by using some coordinates that lie on this function, for example, the coordinate one, two. Substituting 𝑥 equals one and 𝑦 equals two into the equation 𝑦 equals two to the power of 𝑥, we get two equals two to the power of one. This coordinate would lie on the function 𝑦 equals two to the power of 𝑥. Substituting in a different coordinate, for example, the coordinate two, four, we can see that we can verify that this coordinate will also lie on the function 𝑦 equals two to the power 𝑥.

So how do we go from working out the two functions to establishing if two to the power of 𝑥 equals 𝑥 has any solutions or not? Well, we should start by remembering that the solution to an equation is the set of all values that satisfy that equation. Everything on the left-hand side of this equation is found on the function 𝑦 equals two to the power 𝑥. Everything on the right-hand side can be demonstrated by the equation or the function 𝑦 equals 𝑥. The point where these two are equal will be the points of intersection of the functions. But if we look at the graphs, we can see that there are no points of intersection. This means that there are no solutions to the equation two to the power 𝑥 equals 𝑥. And so we can give the answer that the statement “The equation two to the power 𝑥 equals 𝑥 has no solution” is true.