Pre-calculus, how do I find x? Well, let me show you. Here we have an exponential equation with two different bases.
We have 2 and 3 right here, right? And notice that we have 2 to the x minus 5, that's equal to 3 to the x plus 1. Whenever the x is in the exponent, try to take the natural log on both sides. In fact, I'll show you another way to do it after this. But the natural way is, take the natural log on both sides, or any log that you want.
Let's just do natural log, which is the ln. So we have ln of that, which is 2 to the x minus 5, and that will be equal to ln of 3 to the x plus 1. Once we have this now, we can just take the exponent to the front by one of the ln properties. So for the left-hand side, we will have x minus 5 in the front, and because this has two terms, make sure you put on parentheses around it. and then times ln2. And one thing to note is that if you only have one thing instead of the natural log, you don't really have to put down the parentheses.
This is okay. Now for the right-hand side, you do the same thing. Put the exponent to the front. We have x plus 1 times ln3, like that. Now this is just a linear equation because ln2 and ln3, they are just, you know, regular numbers.
zero point something zero point something you can one point something actually you can let me know the numbers in the comment down below but we'll keep that as how they are and then just distribute ln 2 times x write it down as x first and then ln 2 because this will be clearly show that x is multiplied with a number rather than ln 2x this looks like x is instead of the natural log that's not what we are talking about here all right continue minus 5 times ln2, yeah? And then we do the same thing right here. So we will have x times ln3. And then this times that, which is plus ln3.
Now, as always, we are going to put down all the x's on one side and the number on the other side. I'm going to move this to the other side by subtracting x ln3. And of course, we do them on both sides.
So this and that cancel. Then plus 5 ln 2 plus 5 ln 2. This and that cancel. So we get this and that is just x times ln 2 minus x ln 3. That's equal to this and that.
Just write it down as how they are. 5 ln 2. Now we have x and x here. We can factor that out. So I'm going to write it down as x times parenthesis, well you don't need a dot here, x times ln2 minus ln3, that's equal to ln3 plus 5ln2. And finally, we can just divide ln2 minus ln3 on both sides, ln2 minus ln3.
So ladies and gentlemen, the answer will be x, that's equal to, hold that. And this is pretty much it. And of course, you can use a calculator to figure out the approximation, but I will leave that to you.
Now, let me show you guys a prettier way to do this. All right, here we go. Just a prettier way, in my opinion.
We use the rule of exponent first. Notice that we have a subtraction here. And remember the rule of exponent. If we have b to the m over b to the n, what do we do?
Well, we subtract the exponent. top one minus the bottom one so b to the m minus n now can we look at the backwards sure thing so for this right here we can look at this 2 to the x over 2 to the 5 yeah x minus 5 so x on the top here 5 on the bottom now for the addition here remember b to the m times b to the n gives you b to the m plus n when it's a multiplication we add division and subtract so here we can look at this as 3 to the x times 3 to the first just like that now let's work this out 2 to the 5 is just 32 and that's write it down as 1 over 32 times 2 to the x and then right here i would like to just write it down 3 to the x times 3. And just a property earlier, we want to collect all the terms with x on one side and then the number on the other. To achieve that, in this case, let's multiply 32 on both sides, so that this and that can cancel. And then here we have 3 to the x, and this is 2 to the x.
In this case, this is division, sorry, this is multiplication, so we have to divide. Divide this by 3 to the x. Likewise, divide this by 3 to the x, so that this and that cancel. Now... on the right hand side 3 times 32 that is just 96 very nice but for this right here notice they have x and x in the exponent but the bases are different so now another rule of exponent for you guys if we have a over b raised to the m power this right here is the same as a to the m over b to the m when the division we get to distribute the exponent and again can we look at it backwards yes x and x right we can put it on the outside and say that's parentheses 2 over 3 raised to the x power now when we have x in the power again we will have to use logarithm with whatever base that you want the best base for this case is the same base you put that which is 2 over 3. so take log base 2 over 3 on both sides And this and that can cancel.
Yeah. So finally, you can see that x is just equal to log with base 2 over 3 of 96. So of course, this is also a legitimate answer. Yeah.
And then if you enter this on a calculator to do that, I will remind you once again, if you have log base b of x, You will have to use the change of base formula with your calculator. You enter this as log, which is meant to be log base 10 of this number x. And then you divide that by log base b, right?
Because you have the ln key on the calculator, you don't have log base 2 over 3 on the calculator. Unless your calculator has a special button that allows you to enter for any base that you want. in that case you can enter that as well all right your homework let me know the approximation let me know if they are equal or not and a challenge is can you go from here to here expand it but i will leave that to you guys so hopefully this right here helps and that's it