As we all know, yesterday was the day when we wrote the terrifying identities test, and I bet that test left everyone with anything but a smile on their face. As Mr. K said the previous day, it was obvious that there was some anxiety concerning the formidable identities test. But despite all worries and the incredible tension in the room when we were attacking the paper with our minds, I hope everyone managed to perform adequately on the test. Now on for some actual knowledge that was conveyed our way.
Now that the identities test was clearly out of our way, it was inevitable that the following day would bring forth a new unit to our growing intellect. This seemingly excruciating unit is of course exponents and logarithms. But I must add, I do applaud Mr. K's method to introduce this unit, which will be explained in the very next paragraph.
This method of course, all begin with Mr. K up at the Smartboard. He waited there until class had essentially began, then showed us the introductory slide. This slide of course, consisting of numbers to help us segway smoothly into the unit. The slide consisted of these numbers:
Then Mr. K told exactly what we were to do, write each of these numbers in different ways. We were told that we must write each of the above numbers in different forms, but each must consist of an exponent. He started the ball rolling by giving examples of such for the first number, 2. The examples he gave were 4 1/2 and (1/2)-1 . These examples fit as 4 to the exponent half equals 2 since if you have an exponent 1/2, the number is first calculated to the power of 1 then the square of that number. In this case, 4 to the exponent 1 is 4, then the square root of 4 is 2. For 1/2 to the exponent -1, negative exponents, as we have learned in the past, mean we must take the reciprocal of the base, which in the case of 1/2 is 2. After he gave us these examples, he gave us time to determine more for each of the number. He said to either come up with four different ways or as many as we can for each. Then he gave us a while to work on them (although it felt like more than a while =p).
Once Mr. K stopped us, we began to convey all of our thoughts onto the Smartboard. We collectively assembled many different methods for writing each of these, and through writing these we noticed some very obvious patterns. Here are the slides of which our answers are integrated upon:
Just some of the possibilities of writing 2 exponentially. From writing these possibilities, we began to uncover a pattern to determine how to write a number in different ways using exponents. This pattern can be seen through the 81/3 which is equivalent to (1/8)-1/3. Using such examples can show that when we write any number to an exponent x, we can also write the reciprocal of that number to the exponent negative x.
At this point, Mr. K asked if there were any ways we could come up with numbers with exponents that do not include the number 1. For exponents which do not contain 1, such as the exponent 2/3, this simply means that we must square both the numerator and denominator (if the base is a fraction) then take the cube root of both. For the exponent -2/3, we simply take the reciprocal of the base then apply the exponent 2/3.
At last, we wrote down some possibilities for the number 1/4. Another pattern emerged, concerning the number 1/4, this pattern was shown by the expressions (1/8)2/3, (1/32)2/5. This pattern seems to always have 2 as a numerator in the exponent, but increases by odd intervals for the denominator. As for the denominator of the base, it increases by a certain exponent, then is divided by 2. This pattern along with the other can help us when trying to write two numbers with the same base, which will be described in a little bit.
After we were done with all of those, and now that we knew there are an infinite number of ways to write any number using exponents, we moved on. But all of this was a great preparation for our real lesson for today. Solving the following equations:
2x = 64
272x-1 = 3
3x = 1/27
He taught us that we must solve these by making the bases equal to each other. If you set them equal to each other, since the bases are equal, they must equal the same number in the end, meaning that the exponents must also be equal. So if you set the bases equal, you can determine an unknown exponent since they must be equivalent. This is what we must know for this unit, as this is how we will perform and carry out logarithmic expressions.
Such as for the equation 2x = 64, we can solve this numerous ways, but as Mr. K said, and as we have learned before, we must set the bases equal to each other. The two most obvious ways to do so are as follows:
The above also states what to do in each step, although there are bound to be more complex problems in the future, leading to more steps. But each of the above show the basic structure for solving these equations; rewrite the equation with equal bases then you can get rid of the bases and set the exponents equal to each other. But remember, such as in the second example, make sure you keep the x when you alter the exponent if the x is in that exponent. Such as in 64(1/6)x, when the 2x was changed to a power to the base of 64, the x must still remain. Don't just change it to 641/6.
As for the other equations, the same mechanism was used for each solution, so just remember that if the bases are the same, then the exponents must also equal each other.
The work for each equation can be seen here.
In the afternoon class, we didn't learn anything as the afternoon was marked by a workshop concerning racism. In class we just talked about certain events, and Mr. K read out a story of which he was given to conduct the workshop. We began talking about incidents, and certain stories which pertained to the subject, and most of what we talked about in class can probably be found on the podcast. And that's about all I can say for now, as I'm running out of time.
Well that's my scribe everybody! Sorry if it came up late, I've already went back and forth with practicing for talent show and now my parents are angry with me so I have to cut this short. I won't have much time to edit this and even entirely complete it, so I'll be adding more to this post maybe later tonight or tomorrow, sorry for the inconvenience. If there's any mistakes, complaints, or anything whatsoever, just tell me, I still have to edit much of this post. I hope my scribe is useful to anyone that didn't attend class and it's at least sufficient considering the high standards on our blog. See you all tomorrow! OH and one last thing... the scribe for tomorrow will be...
Have a good night everyone! Don't forget to do your homework!