Algebra 2 PBL: Working With Logarithms

                                                     
The fundamental tool needed to appropriately deal with logarithms is the change of base formula. If an equation meets the constraints (a, b, and x are positive real numbers, and neither a nor b is one) then:

The change of base formula does exactly what it sounds like, it allows us to change the base of the logarithm so that we're able to obtain a solution. A good example may be log8^24 (8 being the base). If we were to evaluate that, no whole number "x" (8^x) could create 24. In that case, we can apply the change of base formula, so that it's expressed as log24/log8 which makes the solution approximately 1.53.

So essentially, something like log0.5^9 would be log9/log0.5, which would then equal -3.2.

Once you've gotten the change of base formula mastered, what comes next is easy. Something like log4^20=2x+5 is easy to solve, you just apply the change of base formula to the left side and then simplify from there like you would with a normal equation. 

log4^20=2x+5

log20/log4=2x+5

(log20/log4)-5=2x

(log20/log4-5)/2=x

x=-1.42

For graphing:

Seperate the two equations into y1 and y2, and do what's shown below.

You can also find a solution via graphing, as I show here: 

If you look at the place of intersection, it's at the same mark as x above, so it's verified!

To solve an equation with x on both sides, you do the same thing as you would above:

The solution to problem is approximately 1.87. Also, if there isn't a base for a logarithm such as with log x above, then it is generally accepted to be 10.

And last, but definitely not least, is solving equations with two logarithms! 

log5^(x-7)=-log4^x

The first step to solving what's above is writing a "system of equations", or y1=log5^(x-7) and y2= -log4^x. From there, you'll want to rewrite both equations via the change of base formula: logx-7/log5 and -logx/log4. After the first two steps, there's only graphing and identifying the solutions:

The solution for this system is approximately 7.1.

And that's all for this project! I realize that logarithms expand much farther than this, but for this project I  wanted to cover material mainly from the "Solving Logarithmic Equations Using Technology" time period.

Algebra 2 PBL: History of Math - Vocabulary

1. Abscissa: The term for x-axis in a graph, it's a conjunction of ab (remove) and scindere (tear). The main root is related to the Latin root from which we get the word scissor. The term was coined by Leibniz around 1855.

2. Absolute Value: The word absolute comes from a variant of absolve, close in meaning to free from restrictions or condition. The phrase is known to be first used by Karl Weierstrass when referencing complex numbers.          

3. Acute: is derived from the Latin word acus for needle, with derivatives for that referring to anything sharp or pointy. An acute angle is one that is sharp or pointy.

4. Angle: comes from the Latin root angulus, which is a sharp bend. Ankle comes from the same root.

5. Algebra: comes from an Arabic book that revolutionized how mathematics worked in western cultures. The first word in the book "Al-jebr w'al-mugabalah" became the word Algebra in western languages.

6. Geometry: derived from the conjuction of the Greek word for Earth, Goes, and the term for "to measure", metros.

7. Hypotenuse: comes from the Greek root hypo (for under), and tein (for stretch). The hypotenuse was the line segment "stretched under" the right angle.

8. Multiply: Combines the roots of multi, many, and pli, for folds, as in a number folded on itself many times. Multiply seems to have been first used by Chaucer in "A Treatise on the Astrolabe".

9. Quadratic: is the Latin root for "to make square".

10. Secant: is from the Latin root Secare, meaning to cut. It is a name for the segment that cuts through a circle. The word was introduced by Thomas Fincke in 1583.

11. Subtract: is a conjuction of two roots, sub (under or below) and tract (meaning to pull or carry away).

12. Square: is derived from the Latin phrase Exquadrare, later contracted into its present meaning of the regular quadrilateral.

13. Tangent: Is from the Latin root tangere, meaning to touch, describing two curves which meet at a single point. Tangent is a creation of Thomas Fincke, written by him around 1583.

14. Zero: comes from the Arabic word "sifre".

15. Symmetry: is a conjunction of sum and metros. The prefix refers to things which are alike.

16. Sequence: is from the Latin root sequi, meaning "to follow". In mathematics, it refers to a series of terms in order, so a pattern between numbers.

17. Plus: comes from the early latin word meaning "more".This word is closely related to the Greek root "poly" (many).

18. Negative numbers: the word for negative was introduced by Brahmagupta, a Hindu mathematician around 600 A.D. The Latin root of today's word is negare, meaning "to deny".

19. Obtuse: is from the Latin formation ob(against) and tundere(to beat). An object, when beaten, becomes blunt, dull, or rounded, and the application to an obtuse angle is in this sense.

20. Fraction: comes from the Latin word frangere, meaning "to break". A fraction represents a broken whole, in this sense.

Geometry PBL: Scale Drawings

Purpose: To explain Scale Drawings to others.

Scale drawings are basically any depiction of a real structure based as a model or "drawing". We use these all the times with maps, figurines, etc. To use scale drawings you must first have a hypothetical scenario where you'd need to do so. Let's say you have a drawing of a door, with a scale of 2in.=3ft (meaning that 2 inches of the drawing portrayed is equivalent to 3 feet of the actual door), and the width of the actual door is 72 inches (1 ft.=12in.). The objective is to find the width of the drawing, this can be simplified to: 72 inches= 6ft, 6ft= 4in.(because of the "scale factor" which states that 2 inches on the paper is equivalent to 3ft of the actual door). We now know that the width of the drawing is 4 inches!

More examples could be:

1. A model ship is built to a scale 1cm: 4m. The length of the model is 40 cm. What is the length of the actual ship? For this one it's quite simple, really. The length of the model is only 40 cm, 1cm translates to 4m, so you would multiply 4mx40cm which would give the size of the real ship. The answer for this one is 160.

2. If 1 inch represents 75 miles on a map, then how many inches will represent 1500 miles?
Well, for this one: 1in.=75 mi. so you should take 1500/75 to get your answer. This is because dividing 1500 by the inch conversion of 75 will give you the number of inches the map represents.


3. On a map the distance from Srpingfield to Pleasantville is 6 in. The map scale is 1/2 in. = 20 mi. Find the actual distance between Springfield and Pleasantville. With 6 in. and the scale being 1/2 in., that means there are 12 "1/2's", so multiplying 20x12=240. So the answer is 240 miles.


The following photo is used for reference to scales, and I took it from here.

Blogging Project

So today I decided to do a bunch of PBL's for school, because at least one is required for most courses, and it's worth 5% of your grade. For this PBL, I have to state why taking this "Computer Literacy" course is going to help me get a job. Well:

One factor would be that we have been practicing typing a bit throughout this course. Many jobs, especially technical ones, require a fast typing speed. I took a typing test recently and got 81 wpm, but on the first one I took for this course I got 69, so my speed has definitely improved. Now that I have a faster typing speed, I'm more likely to get a job, at least in the technical fields.

Another reason taking this course may help me get a job is the requirement of a formal writing style. I don't exactly have to use this on my blog, because it's a blog. However, when contacting someone via email or even text, you want to look as professional as possible. This is a very important skill for many jobs, and this course has covered that.

There are more ways in which this class may help me attain a job, but the project said only two paragraphs, so I'm stopping here.

-Daron

Gamification's Awesome!

This is another post I was required to make for my Computer Literacy project. However, I learned something really cool in the process. Apparently, Gamification is the process by which teachers use game design and game theory to make learning more appealing. I found this really interesting, as I'm a big gamer myself, and it's now becoming a learning tool for students. Earning badges and trophies offer positive reinforcement, and virtual identities allow for self-expression, which both offer great incentive. If you'd like to know more on this topic, here's where I got the information from: Information.

Computer Literacy Project

   
1. In this class, I've learned how to modify the format and font styles of pages.

2. I've also learned formal definitions for terms such as RSS, url, etc.

3. Lastly, I've learned how to use google calendar, which seems the most useful for me personally.








                                                                                                          Google Calendar Example^