Surprise! This week's fun thing is worth extra credit! I'm trying to see if anyone even looks at these, so if you do, here's an opportunity for 1 (1!) point of extra credit:
Hey! I have never been able to exactly put my finger on the reason, but students often have trouble with this particular topic. I encourage you to spend as much time on this as necessary for you to have a firm grasp on logarithms. For those of you that will be continuing in math, logarithms will keep coming up. For those of you that won't be continuing in math, knowledge of what a logarithm is can still be important, particularly when interpreting certain numbers and graphs that you will definitely see in your personal and professional lives. In either case, PLEASE be patient with yourself while you gradually learn this topic. If you find yourself getting frustrated, take a break and come back to it later. I am going to provide all sorts of resources for this one because, well, the more the merrier.
Here is your weekly checklist.
Last week the last thing we said was that a logarithm is an exponent. Logarithm is also a 4 syllable word that intimidates a lot of students. I hope to be able to convince you that logarithms are a pretty simple concept and also a completely necessary part of mathematics. Let's talk about where logarithms come from.
Using Logarithm Tables
Logarithms from a exponential/function perspective
Logarithms provide a solution for several very important limitations we have in math: How can we multiply very large numbers together?; How can we express non-integer (or irrational) exponents exactly?; How can we represent vastly different numbers graphically? When we start to look at applications of logarithms, you'll see that there are quite a few commonly used systems of measurement that are based on a logarithmic scale. For now, it is important that we can manipulate logarithms to write them as equivalent exponential equations. Doing so can also help us solve for unknown values.
Basic logarithms
Calculating basic logarithms and some more complicated ones
The "logarithm loop trick", which is absolutely not an official thing, it's just a Gribble thing (see: missy elliot method, uma thurman method), is super useful for being able to convert from one form to the next. We need to convert from one form to the next in many cases depending on where the unknown variable might be.
There are certain "special" logarithms that we use all the time in the real world. Sometimes they are used to simplify fairly complicated exponential equations, and other times they are used to define the scales by which we measure things. Your understanding of these two special logarithms will be extremely important moving forward with the subject.
Discussion of the common log and the natural log
Solving logarithmic equations
So common logarithms (logarithms of base 10) and natural logarithms (logarithms of base e) are used all the time, but their properties are still the same as any other logarithm. We know that if we see a something like "log base b of 1", no matter what that base "b" is, the value of the logarithm is 0. The similarly know that if we see something like "log base b of b", the value of that logarithm is just 1. I can't overemphasize how important both of these things are.
If we know "the logarithm loop trick", there are just a handful of other things we need to understand about logarithms in order to handle basically any problem involving them. Of those other things, three major ones are the product rule, the quotient rule, and the power rule for logarithms, which are the same rules that we use for exponents (remember, logarithms and exponents are the same!)
Using the different rules for logarithms can let us simplify very ugly looking problems (usually they are ugly due to the argument of the logarithm) and make them manageable. There's just one thing we're missing, and it's been mentioned a few times: if we don't have a fancy calculator that can find logarithms of bases other than 10 or e, how can we go about evaluating problems that involve different bases, like, how can we evaluate this on a calculator: 2˟=3 ? (We could graph y=2˟ and y=3 then find where they intersect, but how can we use logarithms to find the answer directly?) The answer is something called the change base formula, and while the formula itself is very important (especially for those of us who don't have calculators that can evaluate logarithms of any base), the process by which we derive the change base formula is the final piece of the puzzle when it comes to being able to solve any logarithmic equation.
Change of base formula
Solving logarithmic equations
We can now solve ANY problem involving logarithms. There are not really any other techniques involved when solving logarithms that we haven't mentioned, but sometimes we will need to apply this techniques in interesting ways.
As is the case with most math, simply knowing how to solve for the unknown in an equation isn't really the gratifying bit. Rather, if we can apply what we know to unique problems and answer legitimate questions, then we can be sure that we have truly mastered the material. So, here are some applications of logarithms.
Why logarithms are useful, and using logarithms to solve application problems.
Using logarithms to solve a novel problem.
Fascinating, right!?
Hopefully now that you've taken ample time to watch the videos above and complete some of the practice problems, you've got a decent grasp on what logarithms are and how to use them to solve problems. To show me what you know, I want you to complete the this google forms activity. Please try to do your best on this so that I really get a sense of what you know! You will also have a quiz (via Microsoft Forms most likely) on this material combined with what we've learned in previous weeks about exponents. I will update you when that is ready, but it will not be due until late next week, because I want you to have plenty of time to review this material.
HERE IS THE QUIZ. Due Friday 5/8 by 11:59PM