If you remember the rules of exponents and keep in mind that exponential functions and logarithmic functions are inverses, then it should be intuitive to you that logarithms follow the same types of rules as exponents. Notice that multiplication and addition are paired up with logs in the same ways that they're paired up for exponents. The same for division and subtraction. The power rule for logarithms states that an exponent can be pulled out in front of the logarithm before evaluating the logarithm. See the example. You can see how this will help us solve exponential equations such as 2^x=5 later, but for now, let's practice applying the properties of logarithms to expand and contract logarithmic expressions. Try the following problems and watch the rest of the video for the full solution. Finally, here is a video on everything you need to know for logarithms just in case you've missed a few things or want another general overview of topics covered so far. It also goes a little into our next section (second half of the video) if you want to get ahead! :) When solving exponential equations, we can sometimes rewrite both sides of the equation with the same base and set the exponents equal to each other. These types of equations are not very challenging to solve, but most of the time we won't be able to rewrite both sides with the same base. For example, what about 5^x = 26? We can't really rewrite 26 with a base of 5.... The idea is that we need a way to get x out of the exponent so that we can solve for it. In order to do this, we need the inverse operation of exponential functions (Remember the inverse operation "undoes" the operation already there. For example, adding and subtracting are inverse operations because +x and -x undo each other. In the same way, multiplication and division are inverse operations because they undo each other.) The inverse of the exponential function is the logarithmic function. Recall that the graphs of two inverse functions are symmetric with respect to y = x. The graph on the left shows 3 common exponential function and their inverses. Notice their symmetry and common characteristics. Before we can take the inverse operation of each side of 5^x=26 in order to solve it, there are a few things we need to know about logarithms first. We'll spend the next few days discussing very important properties of logarithms before returning to solve this problem. (Note: It is important that you fully understand these properties instead of relying on your calculators for this unit.) Basics of LogarithmsPart of my passion for logarithms most likely comes from my high school Algebra II teacher. He was most definitely the most demanding and strict teacher I had in high school (Think no free days...ever! Even that one time a tree fell on him...he still came and lectured!). Anyway, he went crazy one day yelling at us the definition of logarithm and, of course, it stuck! I still remember it to this day... The solution to a logarithm is the exponent to which a base must be raised to produce a given number. The first thing we need to understand is how to convert between exponential and logarithmic form. Watch the video below to see a few basic examples of how this is done. Click here to practice evaluating logarithmic expressions on Khan Academy's interactive practice website. Then click here to try a few that are more challenging. Because exponents and logarithms are related in this manner, there are some very basic properties of logarithms that we like to see happen in our problems! It's super easy to prove these properties if you translate each to their exponential form! Special LogarithmsThere are some bases for logarithms that are more common than others. These special logarithms have unique notation.
Common Logarithm - a log with base 10 Whenever you see "log" without a specified base, you can assume that it has a base of 10 Example: log 100 = 2 because 10^2 = 100. Natural Logarithm - a log with a base e (Euler's number approximately 2.718281828459...) Whenever you see "ln" this indicates a natural log understood to have a base e. (Why use "ln" instead of "nl?") Example: ln e^2 = 2 because their common bases cancel. These are the same "LOG" and "LN" buttons that you see on your calculator. When you use these buttons, your calculator is assuming common log (base 10) and natural log (base e) respectively. (Side note: This is also what Howard was talking about in his song to Bernadette from earlier posts!) These special logarithms are the most common to encounter in real life situations. For example,
The assignment for 2/21/14 was Page 178 #1-10, 46, 48. Due Monday! Watch the video below which shows how to work two examples on how to solve slightly more complex exponential equations than the ones we did in class today. Copy the examples and write down any questions you have about the solutions. You can write your questions in the comments below or simply bring them with you to class! So you thought we were doing trig, huh? My bad...I have too many logarithm jokes to jump over logs! I haven't forgotten that it's Valentine's Day weekend. <3 Before we get started discussing exponential functions, bases of logarithms, and all of the other awesome stuff in our unit, here's a little idea for the guys of the class. Ladies, you only wish you were this lucky... Whoa, wait! Did you catch that?! Go back to 0:29 in the video.... "I’d be solving exponential equations that use bases not found on your calculator, making it much harder to crack." Hey! Exponential equations! In this unit we're going to finally figure out what Howard is talking about... Oh, and by the way, remember that this is the unit I promised for ACT prep! You asked for it! Let's get to it! Exponential FunctionsBefore we can talk about the wonderful world of logarithms, we need to discuss exponential functions. An exponential function is a function that has its variable in the exponent. One of the most famous examples of exponential growth is the "penny doubled" problem. It goes like this: Would you rather have a penny that gets doubled every day for a month or a million dollars? Of course, it sounds like a trick question, but could doubling your money daily when you only start off with a penny actually end up being more than a million dollars in only a month's time? The guys at 1500 Days to Freedom have written a nice solution of a similar problem here. It's short and to the point, so don't be scared to look at it. It yields a really surprising solution! It turns out that you can model the process going on there with the function you see in the graphic above. If you change the 2 to a 3, you then have a tripling function (Please someone come triple my penny a day! I wonder how much I'd have in 30 days...?) Yes, indeed, there's "trouble in the world from all this exponential growth..." In Michael Crichton's book Andromeda Strain, he states:
"The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth." Do you agree or disagree with this statement? Have a response ready in class when we meet next. Also, research at least one other application of exponential functions (research "exponential growth" or "exponential decay") and place it in the comments on this post. Link pages or articles you researched in your response for our class discussion. *Brownie points for relating this to your future profession! Have a super weekend! Don't forget to sign up for Remind101 if you haven't done that already!! The info is at the top of this page if you've forgotten to write it down. See you guys Tuesday! =D |
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