 So in this video I want to show us how to solve some x exponential and logarithmic equations ones that will probably involve Logarithms of some kind like let's say we have the equation Two to the x equals five. Well, if you want to solve this equation You want to get x all by itself, but to get rid of x you got to move the base two to the other side How do you get rid of the base two? Like if I had something like two plus x was equal to five what I would do is I would subtract two from both sides so that they cancel out or If I had instead two x is equal to five I would divide by two on both sides to get x equals five house right yet to perform the inverse operation So what's the inverse operation here? You're gonna be taking the log base two on Both sides of the equation. That's how you get rid of the two And so you're gonna get log base two of two to the x in that situation The log base two and the exponential base two will cancel out and you end up with x is equal to the log base two of Five that is the answer for which we can estimate that with a calculator Don't worry about how that's done necessarily right now, but your calculator could estimate this as two point three two one nine It's gonna be an irrational number But that that's the answer right there for the moment being focus on the exact answer The solution here is the log base two of five. Okay now What if you have the equation x to the fifth is equal to four now? You might be thinking that we use a logarithm here because there is some exponential expression of the left-hand side But the left-hand side where's the variable the variable is the base of the exponential expression not the exponent itself And so this is not really an exponential function. This is a power function, right? We don't want to move the x to the other side. We want to move the five to the other side It's the five that needs to move and so how do you get rid of the fifth power in that situation? If you have x to the fifth is equal to four you don't want to use a logarithm You want to take the fifth root of both sides whoops the fifth root of Both sides thus giving you the solution x equals the fifth root of four for which we can use a calculator again to estimate that solution to get one point three One nine five But the important thing is when you look at these two examples a and b right here the first one we solve using logarithms because the base is a constant and the Exponent is a variable but on on the second example here. We don't use logarithms We use radicals we have to use in this case the fifth root and that's because the base is the variable and the exponent is a Constant the the location of the variable makes a big difference. So when you see something like a to the b right here and You know, this is equal to c. How do you solve an equation like this? Well, the thing is if if in this situation a is constant whoops a is constant and b is the variable in that situation then you're going to use a Logrithm right you're going to use the log base a on the other hand if a is the variable and B is the constant Then in that situation you're going to use the beef root Right, so use radicals versus logarithms. It's an important distinction to remember Now if you have a logarithmic equation, it's usually fairly easy what to do The inverse of a logarithm is going to be the exponential function So if you have log base 3 of x and you want to solve this equation right here You got to move the base 3 from the left hand side to the right hand side And as you move from log base 3 to the other side I'll turn into an exponential So you're going to get x equals something the 8 will be on the right hand side The 3 is on the right hand side. You have to be careful though. This is going to be x equals 3 to the 8th You don't want to by the mistake put 8 cubed right because that would be like oh, I got rid of a radical No, you're not cubing both sides. You're moving the base log base 3 We'll move to the side to become the exponential base 3 so we get 3 to the 8th for which I mean that's 3 times 3 times 3 you're gonna do that 8 times that gives you 6,561 as the result there Again, I use a calculator to help me on that one and then for the next one if you had the natural log of x is equal To 8 how do you solve this one remember the natural log is just log base e Right and so when you move the natural log to the side you're going to switch to the exponential base e So you get x is equal to e to the 8th for which you can consult your calculator and see that that's going to be approximately 2980 point 9580 it's an irrational number. Let's give you four decimal places right there now when you're working base e Most people know to put e as the base because again That's nearly how you always do it, but the confusion comes down here, right? You don't want to say eight-thirds by mistake the order of the exponents matters if you switch the base and the exponent It gives you a different result So try to avoid that make sure that as you move from logarithmic form to exponential form the base is always base 3 You don't switch from base 3 to base 8. It'll be base 3 log base 3 exponential And if you can keep those things straight Then you're gonna be fine solving these logarithmic equations and then also as a reminder with these exponential type Expressions right pay attention to who's the base and who's the variable if the variables the exponent use an exponential That is you'll use the logarithmic inverse if the variable is the base That's a power function. You'll use a radical to solve the equation