Hi, and welcome. My name is Anthony Burrell. And today, we're going to solve exponential equations. And so we're going to see an example where there's a same base number on both sides of the equals sign. And then we'll look at examples where the bases are different.
And this is all going to involve rewriting the base so that we do have a same base. And then we can just use the same strategies for equations with the same base. But before we get into those equations, I'd like to do some review with solving a linear equation where we have the variable on both sides of the equals sign.
So how can we solve for x if 5x plus 3 equals 2x minus 3? Well, our main goal is to group all of our x terms together on one side of the equals sign and then all of our constant terms on the other side. So it doesn't matter which side you choose which terms to be in.
So what I could do is I could subtract 3 from both sides of this equation so that it eliminates the plus 3 here, so I just have 5x. And now I have 2x minus 6, because I had to subtract 3.
Well, now I'd like to get negative 6 all by itself. So I need to subtract 2x from both sides of my equation. So that gives me negative 6 over here but 3x over here because I had to subtract 2x. Now to solve for x, I can just divide by that coefficient. And I see that x equals negative 2.
Well, how does this help me solve exponential equations? So we're going to see an example of an exponential equation with the same base. So 5 raised to the power of x minus 2 equals 5 raised to the power of 2x plus 3.
Now before I get to solving this equation, I do want to make a note that exponential equations can be solved by using logarithms. But for the purpose of these examples, we're going to assume we know nothing about logarithms. And we're going to solve without logs.
Now when we have the same base number on both sides of our equation, what we could do is we could set the exponents. So that expression equal to each other and then solve for our variable. So another way to think about this is that we have 5 raised to some power equals 5 raised to some power. So those two powers must be the same.
So our equation is now x minus 2 equals 2x plus 3. So that's why we went through that review earlier, because this an equation that looks very similar. So to solve for x, we want to have all of our x terms on one side and all of our constant terms on the other. So here what I've have done is I have subtracted 3 from both sides of my equation. So it eliminates the positive 3 here.
But I have x minus 5 now, because I subtracted 3. Now what I'm going to do is subtract x to get negative 5 by itself. But 2x minus x is x. So I've solved for my x value, that is negative 5. So if I were to plug in negative 5 in for x, this would be a true statement.
Now what if we have different bases on each side of our equals sign, so for example, we have three raised to the power of 3x plus 2 equals 9 raised to the power of x minus 5? I can't simply set these exponents equal to each other, because their bases are not the same.
So what I want to do is I want to see if I can rewrite one of the bases so that they do have a same base. So this isn't always possible, keep that in mind. And that's where you would want to use logarithms then if this isn't possible.
But what you do to see if you can rewrite a base is you look at the smaller numbers. So that would be 3 in this case. And then you start thinking about powers of 3 and see if that would be this base. So powers of 3 would be 3, that 3 to the first power. 3 squared is 9, 3 cubed is 27, 3 to the fourth is 81, so on.
I have already said 9 is 3 squared. So I'm going to use the fact that 9 equals 3 squared to rewrite 9 to the power of x minus 5. That would be 3 squared to the power of x minus 5.
So now that I've rewritten this so that I have a common base, I need to apply a certain property of exponents to simplify this side of the equation. And that's going to be our power property where if we have a base number raised to an exponent raised to an exponent again, what we do is just multiply those two exponents.
So to simplify this side of the equation, I'm going to distribute the 2 into x minus 5. So this becomes our base 3 raised to the power of 2x minus 10. Now I have an equation that has the same base on both sides of our equal sign. So I can treat it like our examples from before.
So I'm going to set our exponents equal to each other. So 3x plus 2 equals 2x minus 10. How can we solve this equation? Well I'm going to add 10 to both sides of the equation to undo the minus 10. So I have 3x plus 12 equals 2x.
Then I'm going to subtract 3x. So 12 equals negative x. And if 12 equals negative x, I can multiply or divide by negative 1 to show that negative 12 equals x.
So let's review our lesson on solving exponential equations. Well, if we have the same base on both sides of the equals sign, we can just set those exponents equal to each other. But if we have different bases, we're going to see if we can rewrite the base. So it might be possible to take this a value and write it with a base b raised to some power.
And then we'd have to apply then the power property and just multiply those two exponents. So you can think of this example being solving for nx equals y. So thanks for watching this tutorial on solving exponential equations. Hope to see you next time.