Three guys rent a hotel room for the night. When they get to the hotel they pay the $30 fee, then go up to their room. Soon the bellhop brings up their bags and gives the lawyers back $5 because the hotel was having a special discount that weekend. So the three lawyers decide to each keep one of the $5 dollars and to give the bellhop a $2 tip. However, when they sat down to tally up their expenses for the weekend they could not explain the following details:
Each one of them had originally paid $$10 (towards the initial $30), then each got back $1 which meant that they each paid $9. Then they gave the bellhop a $2 tip. HOWEVER, 3 • $9 + $2 = $29.
The guys couldn't figure out what happened to the other dollar. After all, the three paid out $30 but could only account for $29.
Can you determine what happened?
Answer at the bottom of the page.
Click here for your weekly checklist. As a heads up, the material for this week is really a lot about practice. It is very important to be proficient with exponents or else it will be hopeless going into next week's topic. Consequently, there will be more practice problems assigned and the optional homeworks will be more heavily weighted (worth more points) this week.
We are starting on a major unit that focuses on two main topics: exponents and logarithms. My hope is that at the end of it, you'll be fluent enough with both of them to realize that they are, in fact, the same thing. This can be a really difficult idea to grasp because they look extremely different. Logarithms are especially infamous among high school students in my experience, and I wish I could say the biggest obstacle is the notation. I'll attempt to bring in some explanations of concept from around the internet that I've found to be really helpful when trying to understand both topics. Starting off with some of the basics of exponents...
Once we have a solid understanding of the basic exponent rules, we can begin to examine some more interesting uses for exponents. A question I usually welcome but math teachers often loathe is "when will I ever need to know this?" Well, our present situation (April, 2020) does a lot to answer that question when it comes to exponential functions. There are many applications for exponential function, but in the end they all boil down to either exponential growth, or exponential decay. We can set up a very basic exponential function and learn a bit about how they behave and what their limitations are.
So OK, exponential functions can't have negative bases; if the base is greater than 1 then it is representing "growth", or the function is increasing for larger values of x; and if the base is less than 1 then it is representing "decay", or the function is decreasing for larger values of x. We could actually pretty easily use limit notation to describe the behavior of exponential functions! We won't worry about that right now, though. Instead, the best way to explore how exponential functions work is through applications.
One of the most common uses of exponential functions is interest. A firm understanding of how interest works can be very beneficial when it comes to saving and investing money in the future, or when applying for things like loans and credit cards. Compound interest specifically is a very powerful tool that can be used to earn money (both by you and by those who are trying to earn money from you!)
When we have a standard exponential function f(x)=b˟ and b > 1, we have a basic exponential growth model. Interest works this way, as does the initial growth of living populations like bacteria, deer, rabbits, and even people. However, there are two scenarios where we can model exponential decay, or the exponential decrease of a population or quantity. That occurs when we have 0 < b < 1. For example, if we had a number like (1/2) and we raised to increasing exponents, the value would get smaller and smaller. This can also happen if we have b > 1 but use a negative exponent. From a transformation perspective, a negative exponent in f(x)=b˟ would result in a horizontal flip to the graph. We can think of this also in terms of exponents. We know already that negative exponents indicate reciprocals (recall that xˉ¹=1/x), so if we had a number larger than 1, like 2, and we raised it to a negative power, it would be the same thing as writing 1/2, a number less than 1, raised to whichever power we specified.
There can also be a situation in exponential functions where our knowledge of exponents is needs to extended into the realm of rational exponents in order to solve. This happens when we try to determine things like what interest rate will allow us to earn a certain amount of money in a specified time. We can also run into issues with applications of exponential functions that we simply cannot solve using our knowledge of exponents, and instead need to resort to graphing (or using a detailed table of values) in order to solve.
Next week we will start to talk about how we go about writing exact values for exponents that we cannot otherwise figure out. For example, if I wanted to solve for x in an equation like 2˟=3, how would I go about doing that?
Complete this activity on google forms by Monday, April 27th at 11:59PM so I can get sense of how well you're following along with the content from this week. (Mandatory)
They just did the math wrong. They spent a total of $27 and collectively received $3 back, which totals the original $30.