Logarithms & Exponents: Rules & Laws | Class #07 | A2 Mathematics 9709

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Okay. So for today's class we have logs and exponential exponents. They both are related to one another. Exponential functions you have seen them before. Exponential functions are when your variable is in the power. For example 5^x 10^ x² y = to 5^ x and something like this. These are called exponential functions. Why? Because you can see that your variable is in the power exponential functions because you can see the variable x is in the power. Exponent is power. And you have seen how to solve exponential equations in your O levelc. For example, if you have 2^ x + 2 = to 8, which they're asking you to solve.

Now, this is an example of an exponential equation because you can see the again the variable is in the power exponential equation. This is an exponential equation. You can see the variables in the power and you have some we have done some rules of exponents in your IGCC and O level. Those will be carry forward over here. And to solve those equation, what do you have to do? So the rule was that if you make the bases same on either side of the equation 2 power x 2 is the base and x is the power. So on either side of the equation if you can make the bases same then you can equate the powers. So 2^ x + 2 you can't do anything to this

one. You can do this thing you can write 8 as a power of two. 2 cube is your eight. So now you can see the bases are same. Two and two cancel out. And then you can equate the powers. So x + 2 is equ= to 3. And so your answer comes out as 1. This is your answer. This is how you solve exponential equations. And this is this is something you have done in your IGCSE and O level very much in detail in your as maybe also sometimes this is applicable. Then where why do the logs come in? For example, now if they ask you to solve again look at this example. that you have 5^ x is equals to 3. Now how will you solve this? This is a

sort of a you can say exponential equation because you can see that the power is a variable. The thing that you want to find out but you cannot find it out using the rules that you have done of indices and indices we have learned that if your bases are same on either side then you can cancel out the base and equate the power. Over here you cannot make you don't know what power of five will it give you three right you cannot write this this three number three as a power of five you don't know that so this is where this is where logs come in this is where where logs help. So first of all you have to understand

how do you convert your exponential equation to a logarithmic equation and a logarithmic equation into an exponential equation logs to exponent and exponent to logs. So a generalized format let's suppose it is a power x is equals to b or a power b is equals to c. This is a sort of a you can know this is a exponential equation and if you want to convert it into logs what do you do? This is called the base. This is the power and this is the answer. Right? Now this is your base. This is the power and this is your answer. And if you convert it into logarithmic form, this changes into the base becomes the base

of the log. Log is log g log and the base is still the base a. The power comes outside the equals to power comes over here outside is equals to and your answer c comes inside the log like this. So this is how you can convert your exponential to log and from log to exponential whatever the need be. Right now in this example we want the answer for x 5 power x= to 3 and this is an exponential function. So using the rule over here if we convert this 5^ x = to 3 this would become log and the base will still be the base. So five is the base. The power comes outside this equals to x and the answer of the is after this

equals to part. It comes inside the log. This becomes log 53. Log with a base of 5 and 3. So x answer is log base 53. So you can write it down log base 5 and 3. And you can find it out. Look 0 682. This is your answer. 683. This is your answer. This is how the logs can help you solve exponential function. Sometimes your logarithmic equation cannot be simplified. Cannot be solve cannot be used to further solve the answer. So you convert it into exponential form. So these two formats they are interconnect. They are linked to each other. So according to the need we convert them from this to this or

from this to that. This is where the logs are coming. For example, let me give you another one. If they ask you to solve 10^ x + 2 = to 20. Now again you cannot write uh 20 as a power of 10. So you will convert it into logarithmic form. So log with the base of 10 because the base in the upper equation is again 10 and after the equal to is written 20. So that comes inside the log and whatever the power is x + 2 comes outside the x + 2. And now to find the value of x the two would go with n subtract. So log 10 20 - 2 and that would give you the answer log 10 20 - 2. So 0.6 6 99. This is your answer. Any questions till

now? >> Nice sir. >> H good. Now we will learn more about logs because you have done in detail the exponential. So I don't need to I don't I don't think that I need to explain or write down all of the rules of the exponent. You know the bases are same and when they're multiplying the powers add when the bases are same and they are multiply and when they're dividing the powers subtract power to the power then the powers multiply. Those are the rules that I know that you people know about it. If you don't then let me know I'll send you a recording or something. That is the simple part. You just need a few

revision on the little revision and you will get it. The important part is learning the rules of logs. So now you know yes we will do I was talking about the exponential rule bases we will discuss the log rules right >> now there are three types of logs basics types of for example types of logs uh with respect to base base of log so if your base of the log is 10 log and base of 10. So you can write it down. You don't need to necessarily write log of 10. You write it down simply LG or simply log. These three means the same. So if they have just written down LG or if they have just written down log log without

any base, this means that the base is 10. It's the same thing. Then another one is log of e base e is a constant special constant irrational constant just like pi e is like e and the value of e look it is an irrational number 2.71 8281 you don't need to memorize this but this is just something uh rational number irrational number and if the base is e then you write it down ln lg l lo l lo l lo l lo l lo l lo l lo l lo l lo l lo lg if the base is e then you write it down short form ln ln is also called the natural log so don't be startled if if you see ln this is natural log and the third one is

the the regular log if you have log and if you have written anything over here two or five or three if there's any other base other than this 10 or e they would write it down over here like this if it is log five or if it is log seven or if it is log 12 or anything else they would write it down the base otherwise if the base is 10 they would write lg or log if the base is e then they would write ln they would not write the complete log g they would just write this one so these are the three types of logs. The rules are all the same. You can apply all of the rules that you will learn right now to

any of them. Okay. So now now there are these rules here. I have written down the rules of exponent. You have seen this before. I know a power n multiply by a power m the powers add up n + m. If they're dividing then the power subtract the power to the power is zero. Anything to the power of zero is one. And when you have a negative power you can just do reciprocal and your power sign becomes positive. The sign changes and this is the one which we just revised that if solving exponential equation if your bases are same then you can equate the powers. And here's the example which I found. Look, I converted the four into

two square and then the bases cancel out and I found the value of x. And now these are the rules that you have to follow that you have to learn. Memorize not it's okay. You will eventually after some practice question you will eventually see that you have memorized it. No place here. So look at the first one. So you can see that the bases are same. So there are two logs with the same base A and A and they are adding. So when this happens then you can multiply the insides B and C they multiply. The the the base still remains A but the insides of the log they multiply. Then the second look at as an example

base was 3 three and then you have two over here and seven over here. So the inside it would become 14. Now insides are multiply then they are divide. If they are subtracting if the logs are subtracting with the same bases then the insides would divide but then you have if the bases and the inside are same of the log then it is simply one. So look log 33 3 will be 1 log 44 will be 1 log 70 will be one. Similarly ln e will also be equal to one because I told you ln means the base is e and the inside is also e. So if the base is e and the inside is e so answer would be one. Similarly lg 10 this would also

come out as one because of the same reason the base of lg we know it is 10. and inside of the log is again 10. So log 10 10 would be one. So anything any log with the same base and the same insides would be equal to one. Then this is one of the most important ones the power rule. So you look you have log a and b power m the inside is raised to a certain power inside. So what happened you can bring the power down and multiply. So it was b power m. So the m came down multiply with the with the complete log. So look log 72 power 3. So the three can come down. So this becomes 3 log 72. And similarly you can go up

back again as well. Log 72 cube as well. You can go again to the power as well. It's the same thing. So log [clears throat] 7^ 3 the three can come down. This is the power rule. And similarly if there is something let's suppose it is two over here. So this two will come in the power and multiply with three. So two power six. So it can go both ways. Whatever the rules that you can see over here, they go both ways. Okay? Both ways. Whatever the identity that you learned, they go both ways. According to the situation, you use this uh you use the relevant rules. Then power rule there. There's something that I've written over here.

These two are not equal. Look log a base sorry base is a inside power m. Now the m can come and come come down and uh multiply but in this case the entire log is raised to a certain power. This is something different. Over here the complete log is raised to a power and over here only the inside of the log is raised to a power. So both of these things are different. This is the one where this rule will be applied. The power comes down multiply. Then you have the fifth one. When you have the same, this is this will help you to solve logarithmic equation. When you have the same base on both side of

the equation, you can cancel out the bases and the insides can be equated to each other. So log a n + 2 log a 6 + n. So log a log a cancel out and then you can equate the insides. Copy down these rules for now. This is where I've written special log according to the base. So if the log has the base 10 then you can write it down lg or log and if the log has the base e then you can write ln natural log. And here are some examples to do. and using these rules to help you solve exponent sorry this logarithmic equation. Now you can see how why do you call this logarithmic equation because your variable the thing that you are

trying to find is it is inside the log x is inside the log. So you have to somehow move things around and uh use the rules that we have learned either we convert it into exponential form or use logs to further solve this. There are okay let's take this example log 3 x + 5 is equals to two. Now there are two methods of solving this. One of them is to use exponential form converted into exponential form. So I told you the base of the log and the base of the exponential form it is the same. So base is three. Then whatever is the power of the exponential is formed comes outside the log is equals to log. This is the this

is your power if you can recall it was a b is equals to c and this becomes log a c is equals to b. So whatever was the power was outside there is equals to and similarly I can bring them back in the power as well. So over here outside the log outside is equals to we have two. So this will come in the power of the base sorry 2. So this becomes 2 3 power 2 is equals to x + 5. So this becomes what? 9 = x + 5. And so your x will come out as 9 - 5 would be four. This is it. This is your answer. This is how you solve it. Another method is still using logs. Rules of logs. Now I know one of the rules of logs is that

if I can change if I can have both sides the same log with the same base then I can cancel out the logs. What do I do? I keep the log over here. Log 3 x + 5. And the base of the logs that we are working with it is three. So I what do I do? I I multiply the two with log 3. Isn't it? Because I told you if the base is same and the inside is same, it is equals to one. So it doesn't matter if I write log 3 over here. But why do I write log 3 over here? Because I want to have the same log on both sides. So the two now using the power rule I can do I can bring this two into the power of three. So this becomes log 3²

and over here this becomes log 3 x + 5. And you can see the log bases are same. So log 3 cancel out with this log 3 and you are left with x + 5 = 3 square is 9 and so your x answer is coming out as 4 the same. So look both ways you got the same answer as I converted into this uh exponential form for using still the the rules of logs. Any questions? >> Nope. Okay, I'll do another one. Whatever the rule that you find is what you're more comfortable with, you can go with that. >> Uhhuh. I also felt this converting from log to exponential I used to make mistakes. I preferred this one. I don't know why maybe the same reason as you. I

prefer to do it the using still the logs. So log 2 3x - 1 and five I multiply it with the same log 2 and same inside. So this would become log 2^ 5 log 2 3x - 1. So same base log to log will cancel out e x - 1 is equ= to 32 and so your value of x will come out as 11. This is your answer. You can copy it down the examples and then we'll move on to a different combination. Now using more rules together and try to solve the equation. So question number three part a log 2x plus log 2x - 1 is equ= to log 220. Now I cannot do anything on the right side. On the left side we learned a rule where if the bases are same and they're

adding up the insides would multiply. So log two and the inside would multiply x * x - 1 is equ= to log 2 and so log 2 and log 2 cancel out because same log on both sides and now you can simplify x² - x = to 20 and then x² - x - 20 is equ= to 0 and use your calculator get the answer you will get I think x = 5 and -4, isn't it? 1 - 1 - 22 - 4 and five. Yes, but you will see that you will have only one answers. One answer which this would be the only answer log five positive because this log cannot work for a negative number. You can see if you use a calculator you will get the answer of log 25 which is number but if you but you won't

get an answer log 2 - 4 look it is in uh Iota 2 + 4.5 32 and something at the end you can see I iota so this means this there is no solution no real solution so there is only one answer in this case it is only x= to 5 log doesn't work for a negative number. So this is the only answer and this is how you would solve it. So you use the rule of log simplified method of the log and then it becomes the normal question easy one. Okay, I'll do one more and then you people can try them. I will do question number part D. Part D is 3 + 2 log 2x = to log 2 3 - 10 x okay the three I can write down the three as

3 log 22 because I want to bring the log with it as well so that I can add them simplify them. If I move this to on the power, this would become log 2 x². This equ= to log 2 3 - 10 x. Then this three would also come in the power. So 2 cube. So this becomes log 2 cq. And then the bases are same. So they would multiply. So log 2 2 cq * x² because they they are adding up log with the same base are adding up so the inside will multiply so log 2 3 - 10 x and now you can cancel out the logs logs log cancel log 8 x² = 3 - 10 x 8 x² - + 10 x - 3 is equ= to So your answers are coming out as 8 10 - 3. So x = to 1 - 1.5

and x = to 0.25. So look again your x won't work for a negative number. So you will cancel out this one. This is the only answer. Any questions anyone? Mo all good. >> Thanks sir. All good. >> Good. Good. Copy down this example. This is a good one. You're applying almost all of the important rules over here of blocks. Okay. Okay. So you people have done this one got the right answer as well. Good. Now these are the same things but involving look ln ln but uh I told you they work the same way. So if I let's suppose we take this one two ln 2x + 1 is equ= to 2 ln x + ln 5. And these are very power like

question the power also you see very similar one they're asking you to solve for x. So over here as well if we have the same log on both sides we can equate we can equate the insides. So what can I do? First I bring this 2 to the power because I want to have log only ln 2x + 1² is equals to this would become ln x² + ln 5. And now ln ln they're writing up so the inside will multiply. So 5 x² and this is ln 2x + 1². Now the ln ln can cancel out. So 2x + 1² is equ= to 5 x². And now you'll expand it, simplify, get the answer for x. So 4x² + 4x + 1 is = 5 x². Move everything on one side. So, x² - 4x + 1

1 - 4 1. So, x is coming out as see 0 268 and the other one is 3.732. These are the two answers. So, you use the same rules. There was nothing different. So all of these ones, question number nine, question number 10 as well, the question number 10, they're just asking you to write x y in terms of x. So you have to get rid of the logs and then make y the subject again the same rules. Question number 11 is something interesting. 11 is question number 11 part. Let's do 6 e 2x - 13 e x - 5. Now this is disguised quadratic and after doing the after converting it into a quadratic equation you will solve

it and then replace use the substitution back and get the answer for x. So in this one uh you can see if let your let m any other variable be e power x. So this would lead you to m² being e power 2x. So now you can replace the e power 2x with m² -3 m - 5 is =0 and then you sol you solve the same way for now you will get answers for m. So your m is = to -1 upon 3 and m is = to 2 upon 5 or sorry 2.5 but you need the answers for x not for m. So you substitute the variable back e power x= to -1 / 3 and e power x is equals to 2.5. Now what do you do? One more thing you can either convert it into exponential

sorry logarithmic form get the answer. Another thing that you can do is ln ln on both sides. Taking this is something new. Taking this also helps ln on both sides. This is also which I have used a lot in my time. There are other rules as well that you can rule that you can use to solve this but this is the best one which I use in my time a lot. So ln e power x just like you can cancel out the same ln's on the on on both side you can put in the ln on both sides so ln negative 1 upon 3 now we know this is not solvable because ln doesn't work the logs doesn't work for a negative number so cancels out so ln e power x is equals

to ln 2.5 the power x can come down so x ln e is equals to ln 2.5 and ln e we know it is equals to 1 Same base, same inside. So, x is equals to ln 2.5. Do you get it? Any questions anyone? Okay. So, there this is one more thing that you have learned. You can they can ask you they won't tell you to use substitution but you will see eventually that in some questions substitution will help you to simplify the things before you find the solution. For example, in this equation 5^ x - 1 is equ= to 5^ x - 5 or in the question number seven as well 2^ x. You can just take a substitution. You can replace this two

power x with some other variable. Solve solve solve solve get the answer to that variable and then replace it back. Find the value of x. Explore this thing a bit. Right now I want to do even more difficult or I think the more uh complicated part which is question number 13 in this picture. Okay. Okay. So look in this one answer. So first of all you have your variable in the power of e in the power and then to add the spice they have put in an ln also outside. So first we will have to bring out the e power 2 outside the ln outside the n then we'll have to bring out the x outside the power. So let's see. So what you can do is

first of all you can bring the two over the other side. This would become 1 upon 2 which is be.5. Now to get rid of the ln I give the same base same inside. So 0.5 ln e. So this will become ln 3 - e 2x is equ= to ln e power 0.5. And now you can see the bases are same. Sorry the ln are same. Ln cancel out. So you have 3 - e 2 power x = to e power.5. And now simplify let's see this would become 3 - e power.5 is equ= to e power 2x and now I want to get the x outside the s= to outside the power so I told you take ln on both side now now put in ln yourself now now this will help you now this ln 3 minus e power.5 this is just a constant

You will input in the calculator you will get the answer to this thing. And over here look 2 can come down and you will get l and e l and e will be canceled out. So your x answer will be ln 3 - e power.5 upon 2. This is the answer and then you can use your calculator to convert it into a decimal. The question will not be more difficult than this one. If you understand this thing how the things are moving when to use which and how are we using those so you will easily get this chapter. Okay, >> let's do some questions from the past look similar question. Question number 10 is the same as this one 2 ln 5 minus

e only the difference I think is of the power yes e power minus 2x and over here we have plus 2x otherwise it's the same look at this one question number 11 is a more interesting now you can see the variable that you are trying to find it is in the power so definitely we will have to somehow bring the logs in so that it can bring the power the variable down outside the log. Oh sorry outside the power. So what we can do you have 5 2x - 1 is 2 3 power x you can take ln on both side because you want to bring down the power. So what would this become? 2x - 1 comes down ln 5 and this is inside the log there

are two things multiplying with each other inside the log you have ln2 and you have 3 power x and using the rules of logs backward you can now separate them ln2 plus ln 3 power x isn't it this we have used the first rule of log in the opposite direction so that we can separate the things which are multiplying inside log and now if If you expand this, this would become 2x ln 5 minus ln 5 is equals to ln2 plus the three the x can come down. So x ln3 bring both the x together. So 2x ln 5 - x ln 3 is equ= to ln 2 plus ln 5. I brought this thing on the right side, this thing on the left side. And now take x common. So 2 ln 5 - ln3

is equ= to ln2 + ln 5 x comes out as ln2 + ln5 upon 2 ln 5 - ln3. This is your answer giving your answer correct to three significant figures. So you can use your calculator get the answer. Any questions anyone? Easy or difficult? Practice >> this needs practice. So 1.0 9 this is your answer. Copy this down. We have done almost I think everything every different type of question that can come in logs. We have done substitution. We have done solving logarithmic. How do you multiply divide and cancel out the logs on same side and then solve for the variable. In this one you have to deliberately bring in the

bring in the logs that your power the variable and the power can come down. Take the screenshot or copy this down. And I think this is enough for today's class. We have done a lot of information. I'll share you this worksheet. Do the practice timely.