 Today, I want to talk about how we can solve exponential equations without using logarithms. So if we don't want to solve with logarithms, we can solve exponential equations using the uniqueness property. And what that means is what we can do is write the exponential equations as two exponentials each having the same base. So this property only works in certain cases, but it helps us just to refresh some of our exponent properties and also practice using the uniqueness property. So we'll start with a very basic example, 144 equals 12 to the x. This method, our goal is to rewrite 144 as 12 to some power. So 144 is 12 squared. And so what this says is since the bases are the same, those bases can cancel out. And now we can just set the exponents equal to each other. And so the solution is that x equals 2. Now that's the most basic of an example that we can get. So let's try one that is slightly more difficult. For this next example, our goal will be again to write each of these bases. So one fourth and 64 as a base and an exponent that are the same. So what I would think of right away, I don't like dealing with fractions as bases. So I would rewrite one fourth as four to the negative one. And then we'll keep the two x outside. And then 64, if you think about that, that is going to be four to the third. And if four hadn't worked with 64, we would have had to just try again with a different base. But here it works. And so once the bases are the same, those fours can cancel out. And so here in the exponent, remember when you have a power raised to a power, you'll multiply so that would become negative 2x equals 3. And to solve for x, we'll just divide by negative 2. And so x is equal to negative 1.5. All right, one more example of this method. If you want to pause the video and see if you can come up with a way to rewrite both 27 and 9 as a base and an exponent with the same base. I look at the 27 and 9, and I immediately know the base I want to use is 3. Because 27 is 3 cubed and 9 is 3 squared. So just like the previous example, the 3s now can cancel. And when you have a power raised to a power, you're going to multiply those two together. So on the left, we have 3 times 2x plus 4. And on the right, we have 2 times 4x, which we can just multiply together. On the left, we'll have to distribute the 3, so it becomes 6x plus 12 equals 8x. And then if we move the 6x to the other side, we get 12 equals 2x divided by 2, and x is equal to 6. So those are just a few examples of solving exponential equations using the uniqueness property.