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Exponential Equations. An exponential equation is one in which the variables are in the exponents. Most exponential equations require advanced math to solve and would not appear on the test. The test would only give us equations that could be solved using the basic laws of exponents and roots we have discussed in this module.
So for example, 2 to the x equals 16. That’s an extremely easy exponential equation. This is much easier than you would see on the test. And of course we can solve this when we recognize that 16 is a power of 2. So we’ll just write 16 as 2 to the 4th. And then we can set the exponents equal on both sides because the bases are equal on both sides.
If you see an exponential equation on the test, most likely they will have variables in the exponents on each side of the equation. So what we had there, a variable’s on one side and just a constant on the other side, you probably will not see that. Nevertheless, the insight on the last problem can be broadly generalized.
If two powers with the same base are equal, then the exponents must be equal. So if b to the x equals b to the y, it must be true that x equals y. So this the fundamental idea that we will be using to solve exponential equations. This rule works for all bases other than 0 or plus or minus 1. And the test is not gonna give you an exponential equation with one of those numbers as the base.
Here’s a practice question. Pause the video, and then we’ll talk about this. Okay. So, those bases are already equal. We have a base of 7 on both sides of the equation.
All we have to do is set the exponents equal, and then solve. We’ll add x to both sides, divide by 3. We get x equals 2. It’s good to understand that problem. But still, this is easier than what the test is going to expect you to know about exponential equations.
The test will never hand you an exponential equation in which the two bases are already equal. We’ve seen that last problem, it kind of handed us that on a silver platter. The real test problems are not gonna do that. They will always give you two different bases on the different sides of the equation.
Of course we can’t apply the same slick rule if the two bases are not the same but the test will always give us two bases so that we could change one or both base to make the bases the same on both sides. So what do I mean by this? Let’s go ba, back to that last problem but present it as the test might present it. They might give you something more like this.
If 49 to the x equals 7 to the 6 minus x, then solve for x. So notice that the two bases on, on each side of the equation, they are no longer equal. We have two different bases. But, of course, this still isn’t that bad, we just have to recognize. Of course, that 49 can be expressed as a power of 7.
So I’m gonna start with that equation and I’m going to replace that 49 with 7 squared. And of course I can multiply through the exponent and now this looks like the actual problem we already solved. So in other words, just by doing that one substitution, we were able to make the bases equal.
Now we can set the exponents equal and solve. Here’s another along these lines. Pause the video and work on this. Of course, we have to rewrite that root as a fractional exponent as we learned in the previous lesson.
So the 5th root of 3, we have to rewrite that as 3 to the power of 1 5th. Now, we’ll multiply the exponents. Now, we have equal bases, so we’ll just set the exponents equal multiplied by 5, and then just do ordinary algebra to solve. Sometimes, neither of the basis can be written as a power of the other.
Instead, both bases can be written as a power of some other, smaller number. This is actually the most common scenario in the test. By far, the vast majority of, of exponential equations on the test are precisely this form. Two bases and neither one can be written easily as a power of the other, but both can be written as powers of the third number.
For example, if we had some power of 8 and some power of 16, we can’t write 16 as a power of 8, we can’t write 8 as a power of 16. We would have to begin by recognizing that both 8 and 16 can be rewritten as powers of 2. So we have to rewrite each base as a power of a common, smaller number. And then by using the laws of exponents we can get everything to equal bases and set the exponents equal.
Here’s a practice problem along these lines. So this now is a problem as it might appear on the test. Pause the vide and then we’ll talk about this. Okay.
Well, 27 and 81, we can’t write 27 as a power of 81 or 81 as a power of 27. The first step is to recognize that both 27 and 81 are powers of 3. And we can rewrite them as powers of 3. 27 is 3 to the 3rd. 81 is 3 to the 4th. So we’re just gonna rewrite the equation in terms of powers of 3, multiply the exponents on both sides, well now we have equal bases.
Because the bases are now equal, we can set the exponents equal. And now this just becomes ordinary algebra. We’ll distribute. And we get 2x equals 10. Divide by 2, x equals 5.
And this is the answer. To solve exponential equations, we have to get equal bases on both sides. This may involve expressing the given bases as powers of smaller bases. Once the bases on both sides are equal, we can equate the exponents and solve.
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