Problem

$35-44$ Logarithmic Equations Use the definition …

05:18

Problem 35 Hard Difficulty

$35-44$ Logarithmic Equations Use the definition of the logarithmic function to find $x .$
$${ (a) }\log _{4} x=3 \quad \text { (b) } \log _{10} 0.01=x$$

Answer

$^{*}$ Exponential Form to Logarithmic Form (Vice-Versa)
$y=b^{x} \leftrightarrow \log _{b} y=x$

Related Courses

Algebra

Algebra and Trigonometry

Chapter 4

Exponential and Logarithmic Functions

Section 3

Logarithmic Functions

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Video Transcript

So today we're going to do do some problems related to look at some function. So it's the one off the interesting function in mathematics to look at some function. So, David, we're going to deal these two problems. So if you look at the first problem, um, it's a start log with base four X equals three. Andi, We were trying to find out the value of X on the Beeb. Thought they have given X equals through something expression off log and then this thing we want to solve it. OK, so how we will do that. So before that, we should know about the Logan come function as well as we should know about the exponential function. First we have the basic knowledge about that. Then we can try to solve it. So let's look into the rectum functions. So the guy some function, it's better to explain by example. So if you look at here here, you can see that this example this'll one it say's that law, then with 100 right? So why, It's a look. I've come function because we are using this lot thing, right? So whenever you you you see a function which is using this lock thing. That's a look. I come function so you can see that you have this log on this, then means this stand specified the beast. So this tank specifies the base. So this is the base right on. If you look at this example here, you can see this is the base. So we're Dennis the beast. Right? So it's a lock function with bees. Okay, Now we will look at, um, this different example. So So let's look at the exponential functions or what we mean by the exponential function. So it's something like five days to power three, right? So it's this thing, right? Why we call this as exponential function because we call this five. This five year five is the base on this three is Theo exponents. So this five is the base. And this three is the exponents of dysfunction is known as the exponential function. So you have Toby. You very you know, you should know the technician off. Look, I can function. You should know the definition off the exponential function. If these things are clear to you, then you can easily solve any problem. So for the legate can function. It's simple, like we have a lot function and we have a big kiss right here. We can see that we have the beast 10. And if you look at the exponential function, it's something raised to the power. That power is the exponent on whatever it's to the power that's the base, right? So let's now we look at, um So So there's a B if we can go from the legal term function to the exponential function and we can also go from exponential function when the guys come function. So we will drive to see how we can see. Okay, look. So there is a relation between exponential function and look. I'd come function right so we can convert exponential form to look at the far right. All we can go from the guy's confirm exponential for so there is a vey display is very important. So you have to keep this thing in your mind. So it says that why equals to be exponent eggs right will be Is the beast X is the exponent. This is exponential function, right? So we can see that on the left hand side. We have exponential function, and we want to convert it into a love that can function. Right. So the right inside you can see that how we do that. So this why would be going to apply? We're going to apply log with base. Be right. So this gives you log be with why write This is equals two X. Right? So, um, if you want like, I can give you explain explanation how we are going from this. Why equals to exponential form to look at There is a mathematical reason behind it. Um, so if you want, like, I can give you a review on it, but that we're going to discuss later. So before time, just keep this thing in your mind. Like when you have why equals to be being exponent ex ripe. This is exponential bomb, and we can go from here to locate some form. So how we can do that? It's log B y equals two x. So keep this thing in your mind and we will try to solve the problem. One he got right. So let's look at that one. So let's look at the problem. You remember? It was like we have to find out the rally off X on dhe they have given us like log for X is equal to three. So we are trying to find out the value of X right? So you can see this is my solution. So before that, just look at this part. So dis compared this thing. So if you look at this thing on the left side, it says that log four x equals 23 Just compared the thing that we have started a few minutes ago What? It says that it seemed like log B y equals two X and we know that we can go from this farm. Why equals to be X right? We know this fact right? We have just started this fact few minutes ago. Now just compared this thing, this one. So if you compare these two, you compare these two right so you can see that lot for X equals 23 here long B y equals X. If you compare from here, you can see that this X is equal to three, right? You can see that they are X is equal to three, right? I can see that you have If you compare you can see that, um, X is vanquished. 23 Right. Because this three and the next they are seen right? If you can build, just just make a complete compassion, Right? And we can see that this x and this why is seen. And if you look at the base, this is four. This is B. So you can see that is equal to four, right? It says that the Power X right. So my b is full. Right? So my b is for the power to the exponent three. Right. So this is exponent three. Right? So we are good, right? Good. So this is my wife Ex is my wife. So if you could be a boat side, you conclude that your ex's equals four exponent three, which gives you 64. So we had done right. We have find the value of X. Just you have to keep dusting in your mind. Like if you're wise equals to be X. This is the exponential form. You can go from Tula Guide conform and and the problem we have seen they have given us this look at comfort. So from here. So we go from here through here. Right. So you can go from here. You can go from here to here. You can go from here to here. So it's easy to solve these kind of things, right? On those who are entrusted to know, Like how this this thing is going on. I can explain them surely. Like there is a mathematical, you know, thing behind us. I can solve the car. We are going from this week on the way and how we're going from this form to the another form. Right? Okay. No, let's look at the So we are done. We have computed the valley off ex. Okay, now we are doing the big, but before, just for doing the bad be we will recall some of the properties off the lock. Right? So when you are dealing with long function, you should remember thes three properties, right? So number one is this one. So you can see that if you have love, then it's the piece would be Sten. And you have something e exponent of the barber, right exponent and right. So what eventually happens like the properties is that this m comes here. So this em this m comes out at the front collectives. So it says that this M will be, um, this end coming friends and locked. So you have to keep this thing in your mind. Right? So this is important, right? Okay, Now, if you have now, we will look at this property. So we have locked, then a b. Right. So this is equals locked an a minus log, then be right. So when you are doing the division, take right. So when you are, when you are doing the delusion thing right when you are doing a over B, right? So when you are doing the division thing, right? E overby So then you have loved than e minus log, then be right. So we have Let's even you have this thing that gives you this one, and let's see. Then you have, um, e over be like a multiply by B when you have a over B that gives you the minus sign a lot like the basis same e minus log and B. And when you are doing the multiplication a multiplied by B. Can you get the positive sites? When you're doing the division, you get the negative side which is important. And when you are doing the multiplication, get a positive sign. And when you have the power, that power comes at the front, right? So you have to keep all these three properties in your mind. This is a small recalled like, if you have locked in with black Beast then and you have the same thing here, right? So if you have the same thing here, So, um, just us again. So let's see the same thing here. Um, so you have the same thing. Who? So that gives you four, right? So you have to keep all these things in your mind. All these three property is right. So these are really important, right? So I'm going to do if you have the power, power comes in the front. If you have the division thing over, be then you get the negative sign. If you have a Times B planet in the positive saying, if you had the same base and you have the same thing, whatever you are calculating with that, right, so that gives you what? Right. So you have to keep this thing. Let's look at the, uh, the problem. So if you remember we have this problem, right? So we are going to solve this one. So they are saying if you have logged 10 and they have given zero decimal zero want equals X right. And we are trying to find out value off X. So some of you contain like, Oh, we can use the property off that like, we know how we can go from block function to the exponential function. If you could apply that thing here that I'm not going to work. Right. So if you want to apply here this thing, let's see. Um um, this one. So if you want to apply this thing here, right, if you want to apply this thing that we're not going to work right, because that that will not give you the value off ex. Okay, I can show you how So Let's say you have this thing. We know that how we can go from here to the exponential phone so we can see that from here. Um, this would be like, this is the beast X right. So it's the beast 10. And that is equal to X. Right? So that gives. Use your 0.21 equals 2 10 to the barber X. Right. So just here again. So this gives you if you want to apply that thing, you put 21 He wants to x rayed so you can see that you got a neighbor to solid X from here. Right? So you have to decide, Like better you are able to convert that thing into exponential into exponential form That will give you the value off expert. Not so in the first part, we We tried that wreath that working out, but in this part of the story is not working out. Right. So this is not giving me the value of X. I'm unable to find out the value of X, right? So what I'm trying to do now, this method fails, right? So if you remember the properties that we have learned a few minutes ago that we were going to apply here, so what I did. I know that zero point this thing, I can drive 0.211 over 100 right? I can write this thing. I know how to remove the decimal, right? So I can ride this thing as one over 100 right? Okay, Now what I'm trying to do. I just multiply by then and I divide by death. I did a small trick here. Why? I did. So you've been going through, see? Like why? I did so right. Okay, so before that, just just a few minutes. Please, Just just wait for a few minutes and then even going to see, like, Why? I did so So what I did. I just multi blye. If you look at this expression, if you look at this one like this thing, if you look at this spot won over 100 just multiplied by then. And I divide by death. So we are allowed to do, like, multiply and divide by the same thing, right in mathematics. That's that's all. Directory. AKI. No, Here. If you look at this spot, right. If you look at this one, this left inside here. I used property. It says that if you have something division, something a over B right? We know that we caught the negative sign, right? So this would be love then then So this is here. My is then. And my bree is 100 right? So what I have now So that gives me this love thing. This is Dennis, my A and this gives me 100 But it's 1000. So? So you should remember weaken Dried 1000. As with all three, right? So weakened, right? No, Thursday we can dry 1000 as thing with Bob three. Right. Okay, so we are good, right? Okay. No, If you look at this one, if I do not multiply by Thank you. I will get love then one and I don't know how to solve this one. I don't know the value off this one, right. So I know that I know this fact. Love Ben off. Then it goes to one. So I need this stand here. I need this thing here. That's why I multiplied by then. Here. Right? So that's why I used a small trick. Not going like it's working. So now what I'm trying to do I got this one. So for this part for this spot, right? I've been in use again. The property If you have the power that power, this bomber cops at the front, right? So let's look at here. So that law time then that barbers come in the front. We know that you know that long, then with then gives you one dried No, this fat. So this log bend and gives me one. This is negative. Three. This is one So one minus two. He gives me negative to what? This is the many off X. You're good. So because see, today we will. We have learned how we can go from the guide come function toe exponential function. He's learned about dealer guide can function and the exponential function. And then we see some of the properties off log function how we have to deal with. So just keep doing some problems. If you have any doubts, you can ask me. I'm available here. So did you and Steve. Steve. Good.

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