Divine Algebra Problem | 6ˣ + 4ˣ = 9ˣ

追加: 2026-05-03

[00:00:00]We have this scary looking equation. So, can you find all the real values of X?

[00:00:07]Before we start, let us look at the graph of 9 to the power of X minus 6 raised to the power of X, which will look like this, and then let us plot 4 to the power of X.

[00:00:20]We can see that it intersects at one point only, which is near 1.2.

[00:00:25]But, is it true that it has only one real solution?

[00:00:29]And can we find that point algebraically?

[00:00:32]Okay, so as a first step, let us divide all of these by 4 raised to the power of X. Next, we can use this power rule to get this as 9 by 4 raised to X, this as 6 by 4 to the X, and this as 1. Then we can write this 9 by 4 as 3 by 2 whole square, and then this 6 by 4 as 3 over 2.

[00:00:59]Then we can use this power rule. So, this will become 3 by 2 raised to X whole square minus 3 by 2 to the X equals 1.

[00:01:09]We will now use substitution, where we will call R as 3 by 2 raised to X.

[00:01:16]So, this will become R square minus R equals 1. Now, add 1 by 4 on both sides to get this equals 1 plus 1 by 4 or 5 by 4.

[00:01:28]This is the same as R minus half whole square equals 5 by 4. So, R equals half plus minus root 5 over 2.

[00:01:39]Oh, what a coincidence. It turned out to be the golden ratio. Now, we can again write 3 by 2 to the X as this and this.

[00:01:49]First, let us solve for this negative one. Take log on both sides and note that we will be calling log as natural log, that is with base E. Some people will also refer to it as ln. Now, use this property of logarithms to get X times log of 3 by 2 equals log of this value.

[00:02:11]Note that this value is negative, and therefore it cannot directly represent a valid logarithmic argument, as logarithms of negative numbers are undefined in the real number system.

[00:02:23]So, we will discard this equation.

[00:02:25]By the way, if you are somewhat familiar with complex numbers, then can you figure out the complex value of X using this equation? Let me know your answer in the comments. Great. Now, consider this equation. Again, take log on both sides to get X times log 3 by 2 equals log of this. So, X equals this over this. Now, using this property of log, it will become log of 1 plus root 5 minus log 2. And this will be log of 3 minus log of 2.

[00:03:00]Now, you can leave your answer like this or find its decimal value, which turns out to be 1.187, and that's it. From the graph, we saw that there was only one intersecting point, and it was nearly 1.2, and it turns out to be the same. And also, we proved that we only get one real solution.

[00:03:24]Finally, when we put this equation in an online calculator, we get the exact same result. Isn't that cool?

[00:03:32]So good.

[00:03:40]>> [music]