 In this video, we will discuss how to solve exponential equations by hand. You also can solve them using your graph on your calculator, but I will not discuss that method in this video. So to solve exponential equations, remember that the inverse to an exponential equation is a logarithm. So just starting with a very basic example here. Here we have an exponent with base six. So the inverse to an exponential equation of base six will be a logarithm of base six. So what we want to do to solve this is take the log base six of each side of the equation. And what happens here is because they have the same base, these cancel out the logarithm base six and the base six of the exponent. So all we're left with on the left side of the equation is x and on the right side of the equation we get log base six of 3589. And remember the change of base theorem allows us to evaluate that by taking the log of what's inside of the logarithm divided by the log of the base. And in this case today we will just round to three decimal places for all of these examples. And so the solution here would be x equals 4.568. Let's look at a slightly more difficult example here. So in this example you'll notice we still have an exponential equation, but this time it doesn't start out isolated. So your first step to solving exponential equations is that you want to isolate the exponent. Here I would first add 14 to each side of the equation. And then what you need to make sure of is you can't multiply that four and two together because the four is not raised to the exponent. So next we'll divide each side of the equation by four. And we have the expression two to the x plus three is equal to 4.2. Now that the exponent is isolated now we can solve by taking the log of each side. Because the base of the exponent is two we'll do the log base two of each side of the equation. And on the left the log base two and the two of the exponential cancel out. So we just are left here with x plus three. And then on the right log base two of 4.2 we can evaluate using our change of base theorem. So we end up with x plus three is equal to 2.07. And the final step is just to solve like we would normally solve for x. So if we subtract three from each side of the equation our final answer would be x is equal to negative 0.93. Final example here that we have to do a little bit differently than the others is this expression seven to the x equals four to the 2x minus one. So here we can't just take the log base seven or log base four of each side because they have different bases. So instead you can choose any base that you want to and just take the log of each side. So what I'm going to do here is I'm just going to do the natural log because I know I have something on my calculator that will do that for me. And what I'm going to use here is the power property. So recall that the power property makes it so that the exponents to an expression can drop down in front of the logarithm. So here it doesn't matter that these are different numbers inside of the logarithm but what we can do is drop the exponent in front so that we then have multiplication. So on the left it becomes x times the natural log of seven and on the right it becomes two minus x times the natural log of four. And don't let those natural logs confuse you because the natural log of seven and the natural log of four are just numbers. So I'm going to just type it in here or write it in as a number just to help you see it a little bit better. The rounding might cause it to be off a little bit but natural log of seven is about 1.946 and the natural log of four, I'm going to put it in front here, is about 1.386. So this now is just a standard linear equation to solve. We can distribute the 1.386 on the right and then we'll be able to solve for x. So what I would do next to solve here is just subtract to move all of the x's to one side of the equation and I end up with negative 0.826x equals negative 1.386 and so the final step will be to divide. My final answer in this case is 1.68 is about what it rounds to.