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Solving an Exponential Equation

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Today we're going to talk about solving exponential equations. So we'll start by reviewing how to solve linear equations. And then we'll do some examples solving exponential equations.

So let's start by reviewing how to solve a linear equation. I've got 5x minus 8 equals 3x plus 4. I want to solve this by isolating my x variable using inverse operations.

So I want to start by combining my x variables together. I'm going to do that by subtracting 3x from both sides. That's going to cancel out the 3x that was here. And now I can combine 5x minus 3x, which will give me 2x, and bring down the rest of my terms, minus 8, and then bring down my positive 4.

So now I'm going to isolate my x variable by canceling out this minus 8 or the negative 8. I'm going to do that by adding 8. And I'll do it on both sides of my equations. Here this will cancel. And I'm left with 2x is equal to 4 plus 8, or 12.

Finally, I'm going to divide by 2 to cancel out the 2 that's multiplying by the x. So that will leave me with x is equal to 12 divided by 2, or 6.

So now let's do an example of solving an exponential equation. Here I have 3 to the power of x minus 8 is equal to 3 to the power of 2x plus 1. We know that this is an exponential equation because our variable is in the exponent.

And we know that an equation is a statement of equality, so we know that this side has to have the same value or be equal to this side. And if our bases are the same, they both have the value of 3, then the only way that these two sides can be equal to each other, have the same value, is if our exponents are equal to each other.

So to solve this equation, we can kind of think of our bases being canceled out. We can ignore them, and simply write and solve an equation using our exponents. So we'll have x minus 8 is equal to 2x plus 1.

So solving this equation in the way we just solved our last example, I'm going to start by getting my x variable on one side of the equation. So here I'm going to subtract x or subtract 1x from both sides. So this will cancel.

Make sure you bring down the negative with the 8 is equal to 2x minus x, or minus 1x, is just going to give me 1x, or just x. And then I'll bring down the plus 1.

To get x by itself, I'll then subtract 1 on both sides. Here this will cancel. And I'm left with x is equal to negative 9.

So let's do another example of solving an exponential equation. So I've got 3 to the negative 5x minus 10 is equal to 27 to the x plus 2. So we can solve exponential equations using either logarithms or by rewriting the base using the properties of exponents. And in this lesson, we're going to focus on the strategy of rewriting the base using a property of exponents.

So when we're looking at this equation, in our last example we had both of our bases with the same value. And in this example, that's not the case. One base is 3. And the other base is 27.

So we want to rewrite one of the bases or both of the bases so that they are the same. And we want to start by looking to see if we can just change one of the bases. So we want to see, can we rewrite 27, for example, to be 3 to some power, 3 raised to some power or some exponent.

And we can, because 27 is the same as 3 to the third power. So I'm going to replace 27 with 3 to the third power. Again, because those two values are equivalent, I'm not changing the equation in any way. And then I'll bring down my exponent of x plus 2.

This side of the equation is going to stay the same, because it has a base of 3. And so now I have the same base on both sides of my equation. They're both equal to 3, so as in our last example, we can ignore them.

We can simply write an equation and solve that equation using our exponents. And we need to be careful that this side of the equation now has two parts to the exponents, because we added this three exponent. So on this side of the equation, we'll have negative 5x minus 10. That's my exponent.

And on this side, we'll have 3, the 3 exponent. And then we'll also have being multiplied by x plus 2. We know that we're multiplying because of our property of exponent. Power of a power property says that when you have an exponent raised to another exponent, you multiply them together.

So simplifying this equation, I'm going to distribute my 3 to both terms on the inside. 3 times x will give me 3x. And 3 times 2 will give me 6.

I'm going to get my x variables on one side of the equation. So here I'm going to add 5x on both sides. This will cancel. Again, make sure you bring down here the minus sign with the 10. 3x plus 5x is 8x plus 6.

Now to isolate my x variable, I'm going to subtract 6 from both sides. This will cancel. And I'm left with negative 16 is equal to 8x.

And finally, I'll divide by 8 on both sides. This will cancel. And I'm left with x is equal to negative 2.

So let's go over our key points from today. There are two methods for solving exponential equations. Rewrite one or more of the bases and use properties of exponents to obtain an equation with common bases. Then, write and solve an equation using just the expressions in the exponents. Or use logarithms to cancel out the exponential operation. This requires more extensive properties of logarithms.

So I hope that these key points and examples helped you understand a little bit more about solving exponential equations. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.