Saturday, November 10, 2007

Logarithms: Change of Base Law

Hello fellow students and guests, this is Lina bringing you the scribe post for Friday’s class. In the morning, half the class was missing, but as they say, life goes on and we started the class by solving logarithms.

Let’s remember that, A LOGARITHM IS AN EXPONENT. A very important note that many let slip by them at times. It’s the easiest concept, but one of the most difficult at the same time.

We learned a new law as well, the “change of base law.”

It is a formula that allows you to rewrite a logarithm in terms of logs written with another base. *As defined by my friend, In other words, it allows you to solve an equation that has different bases as shown later on.

It finds for you the exact exponent. When using the change of base law and no base is indicated, it is always “base 10” which is the common logarithm or common law. Since your calculator only allows you to calculate base 10 logs, converting your logarithm into the fraction as seen above automatically changes it to a base 10 log making it easier to calculate. Therefore, it lets you solve any logarithm on your calculator, but keep in mind that you will not always have a calculator for these types of questions. I’ll elaborate on that later.

The reason given to us for why we calculate in Base 10 is simply because we have ten fingers and it is easier to count. If we were aliens with six fingers, then we may just use “base 6”, but let’s not get confused here.

So our first task during the morning class was to solve the following using logarithms.

Our first question:

While keeping in mind from previous lessons that a logarithm is indeed an exponent.

As known in this:
We proceeded to solve the question.

-First, we converted each side in terms of logarithms.

-Also remembering the power law, the exponent “x” is placed in front of the log3

-We then divided both sides by “log3” to isolate the “x” on one side.

-Then we inputted the “log12/log3” on our calculators to get our decimal approximation for “x”.

Our second question was very much the same.

*Even though the brackets aren’t given around the exponent in the original question, you should always put them in for safety measures as not to get confused later on in solving the question.

The answer written in blue is given by a student. Following the same idea as in the previous question and we ended up with the decimal approximation. However, Mr. K showed us a different way of solving the question from the second line in blue where (x+ 1) log3 = log 17.

-From that line, we factored out the (x + 1) out giving us xlog3 + log3 = log17
- We bring the log3 to the right side of the equation
- We divide both sides by log3 isolating the x.
-Badda bing, Badda boom we have our answer in the exact form of: x = (log17 + log3) / (log3)

Usually you would stick to writing the exact answer when you are solving this sort of problem during the non-calculator part of the test or exam. You can write a decimal approximation if you are allowed a calculator, although it’s not necessary unless it specifically states in the question that you have to write out the decimal approximation.

Somewhere in the middle of the lesson Mr. K started talking spontaneously about Base Numbers and how you would make them. For example as seen in the above in Base 2, the symbols used to make up numbers in Base 2 are “0” and “1” only. So the numbers one through four are written using those said symbols only. It all has to do with place value.

I’m not quite sure I can explain this concept well, but here’s a website that you may find interesting that elaborates on this idea.

It’s not that important to know as of now, but you may want to check it out.

By the afternoon class, the missing half of the class joined us once again and we were greeted with yet another pop quiz. My fatal mistake was forgetting for that brief moment that a logarithm is an exponent!

After correcting the quiz together, we proceeded to solve more questions after Mr. K caught the missing half of the class up on what occurred during the morning’s lesson. They caught on quite quickly I must say.

Here’s a sample of one of the questions we worked on. It follows up on the things we learned that morning.

They were solved differently, but are nevertheless the same answer in the end. However, the answer on the left is a little more expanded than the one on the right. You can write it either way as they are both correct.

So somewhere in the middle of both morning and afternoon classes, Mr. K decided to share a joke with us.

We all know the story of Noah’s Ark right? He took two of every animal onto his Ark and landed at the “Mountains of Ararat.” Why he didn’t slap those mosquitoes, I’ll never know. After it docked, he stood at the front of the ship as the animals exited and said, “Go forth and multiply!” And boy those rabbits… Anyway as the snakes were exiting the ship, Noah said once again, “Go forth and multiply!”, but the snakes said they couldn’t. Noah asks why not and the snakes replied that they were “Adder snakes” (get it? It’s a type of snake).

So the second part of the joke. Noah walks to the newly formed forest and after hearing some *crash crash* *smash smash* *bang bang* *hammer hammer*, Noah comes back and once again says to the snakes, “Go forth and multiply!”, but again the snakes look puzzled and say they can’t. Noah then says, “Of course you can because I’ve made you LOG TABLES.”

So there you have it folks, the joke that Mr. K waited three months to tell us. Was it worth the wait or what? Of course I can’t tell it as brilliantly as he did in his very animated voice, but you get the idea.

Well that’s all from me for today. So last, but not least I will name our next scribe. JESSICCA!


- - !! Ecko Montana !! - - said...

Incredible post! This one has to go to the Hall of Fame of scribe posts.

Anonymous said...
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Anonymous said...

Good scribes post Lina. since i was gone for the day i didn't understand the change of base law that well but that clarified things for me. hall of fame worthy.

MSK LEGEND said...

i would like to know wat is "c"? as
log b=log b
a c
log a ?
I would be very thankful to you if you explain me the following question:
pg 52 of ex3.5 book New Additional Mathmeticsby Ho Soo Thong?

dkuropatwa said...

Perhaps you could reproduce the question?