Eksponen dan Logaritma Bagian 1 - Definisi dan Sifat-sifat Eksponen (Kelas X Kurikulum Merdeka)

Assalamualaikum warahmatullahi wabarakatuh. Meet me again, Denny Hendayani on the MATLAB channel. In this video, I will discuss the first material of the 10th grade of the Merdeka curriculum, namely Materi, Exponent, and Logarithma.

But don't worry, for those of you who haven't applied for the new curriculum, you can still learn this material. Because in the 2013 curriculum, it is also taught in interest mathematics. Oh yes, for the material of exponents and logarithms, I will share it in several separate videos and this is the first part of the video. In this first part of the video, we will learn the definition of a divisible number or exponent, as well as the characteristics of exponents.

For those of you who want to record the material I am telling in this video, you can download the PDF form, I will include the link in the description column. And there is an online training, I also include the link in the description column of this video. Okay, let's just discuss the material.

Okay, in this video we will learn the matter of the first exponent and logarithm. The material that we will learn in this video is totally easy, but this is a very important material for you to master because it will support the next material, either in mathematics itself or in other subjects such as biology, physics, and chemistry.

For example, in physics you will learn the intensity of sound, you need the concept of logarithm. And also in chemistry, you will learn the language of oxygen using logarithm too. So make sure you master this material.

Okay, before I discuss the concept of exponents, please pay attention to the following illustration. We are currently in the pandemic, so let you understand better one of the concepts of exponents and logarithms is in virus spread.

For example, there are two people who are positive, infected with the virus. We say that this is the first phase, there are two people who are positive. Well, if each of these two people, each person, transmitted to the other two people in the second phase,

The first person transmitted to two other people and the second person also transmitted to two other people. Well, then in the second phase there are 4 people who are positive to get the virus. Then in the third phase it's the same, each person transmits to two other people. So in the third phase, there are automatically 8 people who are infected with the virus.

And so on until the fourth, fifth, and so on. Well, if we make a table of virus transmission data, in the first phase there were how many people? There are two people, right? In the first phase there are two people, then in the second phase there are four people who are infected.

Then in the third phase, there are 8 people. Can you guess? In the fourth phase, how many people will be infected? How?

Well, if the transmission is still constant, as before, each person transmits to two other people, then from the third phase, there are 8 people, each transmitting to two other people, then 8 times 2, then in the fourth phase there are as many as 16 people who will be transmitted. Then the fifth phase, it means x2 again, right? 32 people. The sixth phase, 64 people, 32 times 2.

Now the question is, can you guess how many people are infected with the virus in the 15th phase? Can you imagine? We have to multiply by 2, multiply by 2, multiply by 2, multiply by 2 until the 15th phase. Or what if the question is like this, in what phase are there 1024 people infected with the virus?

To answer these two questions, we should first understand the pattern of spreading this virus. From 2 to 4, then 8, 16, 32, 64, and so on. This is each x 2, right? Yes or no? x 2, x 2, x 2, and so on.

Now here there is a constant multiplication, a repeated multiplication, 2 times 2, then times 2 times 2. Remember, repeated multiplication, repeated same number multiplication, can be stated in the form of a bracket. For example, here in the first phase there are two people who are separated.

then 2 can be stated as 2/1, where 1 is the phase. For example, in the second phase there are 4 people who are assigned. 4 can be written as 2/2. Then 8 can be stated as 2/3.

Then 16 is 2/4, 32 is 2/5, 64 is 2/6. Now, now you guys pay attention to the part of the bracket. 1, 2, 3, 4, 5, 6, this is the same as the phase, right? The transmission phase. So, if for example the transmission phase is N, then many people who are transmitted are 2/N.

Now we have found the pattern, then we can answer the first question. How many people are infected with the virus in the 15th phase? The pattern is 2/n, where n is the majority of the phase. So in the 15th phase, it is clear that the infected person is 2/15. How many are 2/15?

32,768 people. Easy, right? Now how to answer the second question. In what phase, the number of people infected with the virus is 1024 people? Well, this is the same as we are looking for the phase, which means we are looking for the n, friends. So we can state 2 n = 1024 n. Well, to answer this will be easier if we have mastered the concept of logarithms.

Okay, now you guys pay attention here we use the form of the bracket or the form of the bracket we say as an exponent. Now we will learn the concept of the exponent first. Okay, we start from the definition of a bracket or the definition of an exponent. Surely you guys have learned about the bracket number. Even from elementary school, elementary school and junior high school we have also learned this. So let's repeat it.

The definition of a number, for example, a number n, a number is the form of a multiplication that repeats as many numbers as possible, friends. So if a number n, this means a times a times a times n. So multiplied by n times.

Next, there is a term that you need to know about this problem of the number of digits. Pay attention to the shape of this a-n. A is the number that is ranked, friends.

A is the number that is divided, it is usually called the base number or basis. Once again, remember, if I say the base number or basis, it means the number that is divided. Then n, n is an exponent or a unit. Easy, right? For example,

For example, 3/4 means what is the basis? The basis is clear here is 3 and 4 is the exponent. 3/4 means 3 times 3 times 3 is 4 times.

What is the value? 3 x 3 = 9, 3 x 3 = 27, 3 x 3 = 81. Remember, 3 x 4 = 81. Don't answer 3 x 4 = 12. Be careful. 3 x 4 = 81. Then what if the basis is negative like this? Negative 2 x 3 = the same. It means negative 2 x negative 2 x negative 2 = 3 times.

Negative times negative is positive. Then multiplied by negative again, then the answer is negative. Negative, how much? 8. Then another example, if the basis is division, it's the same. For example, half of the fourth grade, means half multiplied by 4 times. How much? 1/16. Okay, this is the concept of a grade number. Remember, a grade number is the same as the form of a multiplication that repeats as many grades.

Now we will learn the characteristics of the subordinate number or the characteristics of the exponent.

Now we start from the positive positive number first. The first attribute, if you find a form of a multiple of a number that is the same basis, for example, a m times a n, if the basis is the same, then the number can be counted. So a m times a n is the same as a m + n. Remember, the basis must be the same, then we can use this attribute. Example,

We will simplify this form, 3/2 times 3/3, this is the same basis as 3, right? So this value is the same as 3/2 + 3 or 3/5.

Easy right? Now we will try to prove it without using this property. 3/2 times 3/3. 3/2 means 3 times 3 is twice as much as 2 times. Then multiplied by 3/3 is 3 times 3 is 3 times. Now, friends, pay attention. Here there are 3 times 5 times. So this is the same as 3/5. The result is the same, right?

Now we continue to the second attribute. If earlier the divisible number is multiplied by the same basis, the number is added. Now if divided, if there is a divisible number that is the same basis, then the number is reduced. For example, 5/6 divided by 5/2 is the same as 5/6 - 2 or 5/4.

Let's try to prove it. 5/6 means 5 multiplied by 6 times and 5/2 means 5 multiplied by 2 times. Or we can write it like this. Now look at the top and bottom. 5 divided by 5 is 1. So we get 5 x 5 x 5 x 5 = 4 times or 5/4. Proved?

Okay, let's continue to the third attribute. The third attribute, if there is a divisible number, it is divided again, the number can be multiplied. For example, 5 divided by 2 divided by 3, this value is the same as 5 divided by 2 divided by 3. How many times 2 divided by 3? 6. 5 divided by 6. Let's try to prove it.

5/2 divided by 3 is the same as 5/2 times 5/2 times 5/2 times 3 times. While 5/2 itself is 5 times 5, right? So 5 times 5, 5 times 5, 5 times 5 is 3 times. Well, here there are 5 times 6 times. So this is the same as 5/6. The result is the same, so this is proven. This trait is proven.

Let's move on to the next feature. If there is a multiplication then divided, the contents in this box must be divided equally. For example,

For example, 3/2 times 5/3 is divided by 4. So all of these in this column must be divided by 4, friends. 3/2 is divided by 4 and 5/3 is divided by 4. Now there are divisible numbers that are divided again. We multiply this number by this attribute. So this 3/2 times 4 is 8 and this 5/3 times 4 is 12. Easy, huh?

Okay, let's move on to the next attribute. This attribute is almost the same as the previous attribute. If there is a multiplication, all that are in this column are divided by n. Likewise for division, for example, a divided by b is divided by n, then all that are in this column must be divided by n. So a divided by n divided by b divided by n.

For example, you will look for 2/3 divided by 3, this value will be equal to 2/3 divided by 3/3 2/3 is 8, 3/3 is 27, 8/27 That is the exponent for positive-digit numbers. Then what if the number is 0 or negative? Well, there is another attribute, friends.

for a value that is zero, no matter what the basis is, no matter what the value of a is, as long as a is a real value and not zero, then the result is 1 remember, no matter what the basis is, except zero and the basis is real, if it is divided by zero, the value is 1 remember, then for the negative value

If a-n is the same as 1/a^-n with a is real, a is not 0, and n is a round number. Okay, you should note this down, you must remember. Now for the divisible number. For the divisible number,

For the characteristics, it is the same as the positive level that was previously, there are also characteristics that were previously, but there are additions. For example, you find a m/n, the form of the division of the level, this we can change to the form of the root.

So, a^n is equal to the root of n from a^m. This becomes a unit and n becomes the index of the root. We will learn the form of the root in another video.

Okay, that's the definition of the number of units or exponents and their characteristics. Now we will try to discuss some examples of questions. Oh yes, before I continue to discuss some examples of questions, for those of you who want to learn with us with a choice mentor, please join our online bag, MATLAB. To be more clear, please see the Instagram of Bang Soal Matematika or contact us through the following WhatsApp number.

Okay, now let's try to discuss the first example. Determine the value of P, so that the following equation is correct. A, 3/4 is divided by 2, equal to 3/P. Let's try to answer A first.

For this kind of question, make sure you make the left and right squares must be the same base number. Once again, remember to make the base the same. For the A, the base is the same, friends. The base is the same as 3. We just use the exponent property. If there is a number of digits, it is divided again. For example, A is divided by M, divided by N. What is the number? Try it.

We multiply the number by m times n. So, based on this property, we get 3/4 divided by 2, which is 3/4 times 2. How many? 8. Same as the right area, which is 3/P.

Now you see, the left and right corners have the same basis of 3. If the basis is the same, we just have to look at the rank. The left rank is 8 and the right rank is P. So 8 = P automatically. P = 8. So the answer is P = 8. Okay, now we discuss the part B.

The way is the same, we make the left and right spaces have the same basis. And here, coincidentally, the basis is the same as B, we use the exponent property. If there is a multiplication of a unit with the same basis, for example, a to the power of m times a to the power of n, what is the unit? Let's try to sum up the unit. So a to the power of m plus n. Well, based on this property, then in the left space,

B, the rank P times B rank 5 is B rank P plus 5. P plus 5. Same as the right side, B rank 9. Well, this is the basis is the same as B, right? Now let's just look at the rank, friends. Rank on the left is P plus 5, same as the right side is 9.

How many times 5 is multiplied by 9? How many times? Obviously, P is 4, right? Or it could be P is 9 minus 5, so P is equal to 4. This is the value of P for the question in section B. Now we discuss section C. 3P/P = 27P/3.

3P divided by P equals 27P divided by 3. Now for the C part, I will change the right area, friends. We make the right area the same as the left area. The basis on the left is 3P. So we make this 3P too. Well, I change this 27. 27 is the same as 3 divided by 3, right? 3 divided by 3 times.

pi divided by 3. Now we will use this property, if there is a multiplication, a times b, for example, divided by m, each of them in this curve is divided, right? a divided by m times b divided by m. So this part, this is the same three, friends. Well, we can get the number out. So 3pi in the curve, the number is 3. The left section was 3pi divided by p, right?

3P, the rank P. Now, you see, the left and right sides are based on 3P. So, the rank P is equal to 3. Easy, right? Okay, let's continue to the second question.

Okay, now let's try to discuss the second example. Simplify the following shapes. Here are 4 shapes and I start from the first one. 2/4 times 3/6 divided by 2/3 times 3/2 divided by 3. Okay, we will simplify this shape, friends.

To simplify this shape, you can simplify the one in the hole first, or you can also immediately fold it with the board outside.

But I will do what is in the box first, we simplify what is in the box first, friends, pay attention, here is the same basis, 2/4 divided by 2/3, remember the exponent property, if there is a division of a number of levels that are the same basis, for example a^m divided by a^n, what is the level? The level is reduced, right? m - n, then 2/4

divided by 2/3 is 2/4-3. 4-3 is 1, right? For the first level, you can write the level or not. Now for the base 3, 3/6 divided by 3/2, it means 3/6-2.

4. Dibangkatkan lagi. 3. Nah, gunakan sifat eksponen kalau ada perkalian, teman-teman. A.

a times b then divided by all the ones in the dash, you divide it. So a times m times b times m. So all the ones in the dash, you divide by 3. 2 divided by 1 divided by 3, so 2 divided by 1 times 3.

Then, 3/4 is divided by 3 again, so it becomes 3/4 * 3 = 12. Now, if we have solved it like this, the basis is different, so we can't do anything else, friends. So the answer is 2/3 * 3/12. This is quite simple. Okay, now let's try to discuss section B.

a^3 * a^4 then divided by a^7 then divided by 5 I try to simplify this part first because this is the same basis as a a^3 * a^4 the basis is the same multiplied by what is the unit? try the unit we sum 3 + 4 = 7 then divided by 2

then divided by a7 then divided by 5 again. Well, we summarize this section again, a7 is divided by 2.

The number of units is multiplied again, we multiply the units. So, a^7 x 2 = 14 divided by a^7, then multiplied by 5. Now, we simplify this section. a^14 divided by a^7, the basis is the same as a. If divided, we subtract the units.

A-14-7=7, then divided by 5. If the number of digits is divided again, we multiply the digits. So A-7x5=35. This is the simplest form. Easy, right? Let's try to discuss section C.

4A squared divided by 0 is reduced by 3B squared 0 divided by 2. Remember, no matter how much the basis is, except 0, if divided by 0, the value is 1, friends. So, this part here, the value is 1. So, no matter how much A is, if divided by 0, it must be 1. 1 is reduced, but

but in this section 3B zero, friends pay attention that zero is only B, three are not included so 3 times B zero is 1 remember, 4A square zero means the whole basis is this 4A square but different from this 3B zero means the basis is B only, which is divided by zero is only B, it's different then we divide by

2 = 1 - 3 * 1 = 3 / 2 1 - 3 = -2 / 2 = -1 Very easy Okay, now we discuss the D

3M times 3M squared divided by 2 divided by 3/11 times M squared then divided by 3. Okay, as usual, I'll do it first, which is in the curve. This is 3/1, friends. This is 3/1, this is 3/1. It means if multiplied, 3/1 times 3/1 is 3/2, right? 3/2.

Then the M is M1, then the right one is M2. M1 times M2 means M3. Then it is divided into 2. Then the name part is 3/11 times M2. M2, then the whole is divided into 3. Same as

We finish the top part first. 3 x 2 = 2. The number of units is divided by 2. We multiply the units. So 3 x 2 = 4. Then m x 3 = 2. So 3 x 2 = 6. Then the bottom part. 3 x m^2 = 3.

Well, here is the same basis, 3 together 3/4 divided by 3/11 if divided, the number is reduced to 3/4 - 11, how much? -7 then times m m/6 divided by m/2 6 - 2 4 then divided by 3 so we get 3/-7 times 3 -21

then m 4 times 3 = 12. Here there is a negative number, we use the exponent if a negative number n is equal to 1 per a positive number n. So let's make this positive number, we move it down. This becomes 3 times positive number 21. This is already positive if m.

So we have to keep it above, M 12/3 21. This is the simple form. Okay, now let's discuss the third example. Change the following form to the positive form. Let's try A first.

x -1 divided by x -1 + y -1. The property we use, remember -a -n is equal to 1/a +n.

So, x to the negative 1 is the same as 1/x to the 1 or we can write 1/x. Then, minus xy to the negative 1. Well, here the negative y is the x above, the y is changed, so per y to the 1 or per y.

Then the part of the denominator, x to the negative 1 is 1/x, then add y to the negative 1 is equal to 1/y. Clear? For a question like this, it will be easier if you have a smooth operation of division.

If there is a/b, then minus c/d, how is the easiest way? Friends multiply, a times d, ad, then because this is a reduction, minus b times c, bc, then the bottom part you guys multiply, b times d, bd. Then how about this number? Yes, it's the same, it means here we just have to change the number.

From there we get 1 times y is y minus x times x is x squared divided by x times y is xy per the bottom part 1 times y is y plus x times 1 is x per the bottom part xy

Now, to simplify this form, you can multiply the top and bottom parts by xy. So we multiply y-x^2/xy by xy, which means we subtract xy.

So, y-x^2 per this part is also the same. We subtract the xy, so y + x. So this is the simple form, the positive square. We can't simplify it anymore. Okay, now we continue to discuss the part b. For part b, x^-1 times y, we can change it to y per x. Because the negative square is only x. It means x that moves down.

Then minus xy -1, which means x per y. Then the name part, x -1 times y -1 is 1 per xy. Same as we worked like before, y times y is y squared, minus x times xx squared times x times y is xy.

divided by 1/xy. The upper and lower part, the denominator and denominator, are equal per xy. So we just multiply the upper times xy, the lower times xy. So we multiply by y squared minus x squared per xy times xy, we subtract the xy.

So, y squared minus x squared, then here we subtract the xy per 1, or we can write y squared minus x squared. Now this is the positive bracket. Easy, right?

Okay, now let's try to discuss the example of the fourth question. Which one is the largest among the following numbers? We will compare the numbers between A to E. We compare which one is the largest. The easiest way is to change between A to E into a unit number with the same basis. Here the basis will be different. Some have a basis of 8, some have a basis of 4, 16, this is 4, this is 2.

The easiest way is to change it to the same base number. What is the base number? Let's see. Let's make it all 2 bases. 8, 4, 16. We can change this to 2/s. Let's change it to section A. 8 is the same as 2/3. 2/3 divided by 3, then divided by 2.

What is the number of units that are divided by another unit? We multiply the units, then we get 2 units 3 times 3 = 9, 9 times 2 = 18. This is for option A. For option B, we change it to 2.

4 we change to 2/2, then we are divided by 32, then we get the number of units divided by the units again, the units are multiplied by 2/2 times 32, which is 64.

Okay, let's try C. 16 is the same as 2/4, guys. 2/4 then divided by 18. Same as 2/4 times 18. How much? 72.

Then the answer option D, 4/4 is divided by 10, it's the same as 4/4 x 10 = 40. Now we change 4 to 2/2, then divided by 40.

2 times 40 = 80. Now, the E, we don't have to do anything. 2 times 81. Now, friends, look. Between A and E, we have changed it to a unit with a base of 2. Now, we just compare the units. The biggest value is when

The highest rank is the highest rank. Here is rank 18, 64, 72, 80, and 81. Clearly, the biggest is 2 rank 81. Well, this is the highest value. Easy, right?

Okay, this video ends here. For those of you who want to learn PDF form from the material we learned in this video, please download it. The link will be included in the description column. And for those of you who want to practice, there is an online class that we have provided. You can also see the link in the description column of this video. Later, after you try to work on it, you can immediately see the score. If the score is still low, try to study again and work again, okay?

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