 In this video we are going to look at a few examples of solving exponential equations using inspection. So later on we will talk about how to solve exponential equations using logarithms, but first we just want to use some properties of exponents in order to solve these. So the key thing to remember when solving these is you want to be able to write each side of the equation as an exponent using the same base. So in this case on the right side we have just 12 to the z. So that we don't really need to change or you could, but the question to ask is can you write 144 as 12 to some power? And in this case we can, 144 is the same as 12 squared. And so we get 12 to the second equals 12 to the z. And what happens is when the bases are the same they can cancel each other out. And we are just left with the exponents and so z will equal 2. Now let's look at a slightly different example here. So this is an example with a fraction and our goal is to write again both of these bases. So 1 fourth and 64 as an expression with the same base. So you can play around with it a little bit if you want to. But I guess some of this will take a little bit of practice to get used to. One thing to remember is the way that you end up with a fraction as the base is with a negative power. So 1 fourth is the same as 4 to the negative first power. And then 64 is the same as 4 to the third power. So in this case again our bases will cancel. What I like to do before cancelling my base is combine the exponents here at the top. So remember when you have a power raised to a power you'll multiply. So in this case I'm multiplying negative 1 times 2x which is just negative 2x. So what I end up with here the fours will cancel out because they are the same base. And I get negative 2x equals 3 if I solve for x, x equals negative 1.5. Now just one more example to look at. 27 to the 2x plus 4 equals 9 to the 4x. So again try to think of a number that you could use as the base and come up with 27 and 9. So pause if you need to and play around with it. In this case 27 can be thought of as 3 to the third. And then I don't change that to x plus 4 at all. And 9 can be thought of as 3 squared. So just like on the previous I want to combine these exponents. Power to a power means we multiply. So this will become, you have to distribute this 3 to each part. So 6x plus 12. And then on the right we just get 8x. Now since the bases are the same we can cancel. So 6x plus 12 equals 8x. And if we solve that for x we'll subtract 6x from each side. So 12 equals 2x. And if we divide by 2, x equals 6. And of course it's always a good idea to go back to the beginning and plug it in. But this is the concept of solving exponential equations by inspection rather than using logarithms.