This week I only got through the unit Logarithms which was disappointing. I had a somewhat troubling issue come up at work which was fairly distracting and was a major factor why I didn’t get as much done as I would have liked. I also took my dog out every morning for a walk which ate into a lot the time when I’m most productive. Next week I should try and adjust my schedule and commit all larger part if the mornings to working through KA.
Regarding Covid-19, things are currently more or less what I expected them to be this time last week. The USA’s current number of active cases is >300K and it doesn’t look like the curve will be decelerating anytime soon. Canada is currently sitting at ~15K active cases. I think the USA will be at 500K this time next week. If that’s the case, I think the attitude of the general public is going to take a dramatic shift towards further towards fear, anger, worry, etc. My impression is that Canada seems to be doing a pretty good job in keeping our numbers relatively low, in large part due to our social distancing. Personally, I think this is my third week of not working and social distancing. Because I lean towards the introverted side, I feel like I’m likely finding this easier than a lot of people in terms of feeling isolated but, even still, I’m starting to get a bit bored. It would be nice to go out and see people but I think we’re probably a minimum of 6 weeks away from being able to do that. This week I heard a great quote made by Winston Churchill that was something along the lines of, “we haven’t reached the beginning of the end, but we may have reached end of the beginning”. I think that may my accurate to where we’re at now.
Before I started the unit Logarithms, I couldn’t remember if I had ever been taught about logarithms back in school. Now having gone through the unit, however, I’m certain that I had never learned about them. I had a weird sense of pride come over me when I realized I was learning something that I had never learned in school. I found out that logarithms are another way of writing/thinking about exponents. In my mind, I think of them as backwards exponents (I don’t think this is exactly what they are, technically speaking).
- Exponent Form
- a^b = c
- Ex. 2^3 = 8
- Logarithm Form
- Log_a(c) = b
- Log_2(8) = 3
- Notation:
- a = Base (usually the base is denoted with the letter “b”)
- b = Exponent
- c = Argument
In the same way that reading an exponent ‘out-loud’ you’d say, “raising ‘a’ to the power of ‘b’ gives you ‘c’’, the logarithmic form essentially reads as, “the ‘log’ ‘a’ of ‘c’ would equal the exponent ‘b’”. (Again, not sure if this is the correct way of saying/thinking about it, technically speaking.) One thing that I was really happy to learn in this unit was how write a number here (as in on this blog) and indicate that the number should be read as being written below another number. By this I mean that, in the same way that the upwards arrow symbol (“^”) is used to indicate that a number should be read as being written at the top of another number (such as an exponent), the underscore is used to show that a number is written below another number. This means that writing “Log_2” would be written out on paper to show the “2” written at the bottom of “Log”.
I learned about 4 different formulas/rules regarding logarithms in this unit; the Product Rule, the Quotient Rule, the Power Rule, and the Change of Base Rule.
- The Product Rule
- Log_b(M) + Log_b(N) = Log_b(M * N)
- This rule states that adding the arguments of two logarithms with the same base is the same thing as multiplying the arguments together first and then taking that product to the logarithm of the same base.
- The Quotient Rule
- Log_b(M) – Log_b(N) = Log_b(M/N)
- Quite similar to the Product Rule, the Quotient Rule states that Subtracting the arguments of two logarithms with the same base is the same thing as dividing the exponents together first and then taking that quotient to the logarithm of the same base.
- The Product Rule
- Log_b(M^p) = p * Log_b(M)
- This rule states that an argument being raised to a certain power is the same thing as removing that power from the argument and multiplying the entire logarithm by that number.
- Proof:
- The exponential equation a^b = c is the same thing as the logarithmic equation Log_a(c) = b.
- If we take both sides of the exponential equation and raise each side to the power of d we get (a^b)^d = (c)^d which can also be written as a^bd = c^d.
- We can then switch the exponent and the argument to go from the exponential form into the logarithmic form to give us Log_a(c^d) = bd.
- At the beginning of this proof we said that Log_a(c) = b so by replacing b in the equation directly above we’re left with Log_a(c^d) = (Log_a(c)) * d.
- Proof:
- The Change of Base Rule
- Log_b(M) = (Log_x(M))/(Log_x(b))
- This rule states that the value of any logarithm is the same as attaching a logarithm of another random base to both the argument and the initial base (turning the initial base into an argument itself) and diving them by each other.
- Proof:
- The exponential form of Log_2(50) = n would be 2^n = 50.
- Taking the log of both sides of the exponential equation to a random base x gives us Lox_x(2^n) = Log_x(50).
- Using the Product Rule we can then say that n * Log_x(2) = Log_x(50).
- Dividing both sides of the equation by Log_x(2) to isolate n leaves us with n = (Log_x(50))/(Log_x(2)).
- Since at the beginning of this proof we established that Log_2(50) = n then we can say Log_2(50) = n = (Log_x(50))/(Log_x(2)).
- Proof:
One thing I found difficult with this unit was understanding the order of operations when being asked to use a few of these rules/formulas at the same time. I need more practice with these types of questions to get a better feel for the correct order of steps to take to go through them in order to come to the right answer. When I first started doing these questions, I also wasn’t sure how to use the logarithmic functions on my calculator. I figured it out fairly easily although I still find it a bit weird and unintuitive.
I have 5 units remaining in the course Algebra 2. I’d like to get through the course by the end of the month which gives me 25 days to get it done. I think getting through a unit every 5 days is doable for me but I will need to buckle down and focus if I’m going to make that happen. Though it will be somewhat challenging, I think it’s a reasonable and achievable goal and am motivated to do it. If I can get through the next two units this week, Transformations of Functions (0/1000 M.P.) and Equations (0/1000 M.P.), I will be in good shape to get through the course by the end of the month. Time to get to work!