Now that we’ve seen what logarithms are,

let’s look at how they behave and look at the properties of logarithms. The first property I’ll go over, I think you already know , is that for any

base b, the log base b of one is going to be zero. Recall that b to the 0 power

is always one. So of course that follows. Going backwards, the log using base b to

get one the exponent the log is 0. The second property is that for any b, any

base b, the log using base b of b to the X is X. Now, sometimes that’s so easy it’s

hard. That’s like saying who’s buried in

Grant’s tomb or what was Elvis’s first name or when was the War of 1812? You see how this is so easy it’s almost hard. It’s like saying what is the exponent of

B to the X using a base of b. Well, of course it’s x. Now why is that

important, this second property here? Well, this is going to allow us to solve

an equation like this one. Looks difficult but in the future what we’re

going to do… we’re going to introduce a new tool

taking the log of both sides and if we take the log of this (think about it) on

the left side, the log base 2 of 2 to the X is X. It’s going to allow us to solve

some very difficult equations. Now that X will equal the log base 2 of 7 whatever

that is but, and, and we’ll figure that out; we’ll work with that later. So I hope

you see that second property is going to be quite useful, that the log base B of B

to the X simplifies to X. Look familiar? Okay, let’s move on to the other properties. Basically, this is going to be divided

into three properties. This is the one you’re going to use way more often since

logarithms are exponents. Remember the rule of exponents. When you

multiply expressions that contain exponents you add the exponents. Well, guess what? Logarithms are exponents so

when you multiply an expression that contains a log you add the logs. Another property of logarithms since

logarithms are exponents is that when you divide expressions with exponents

you subtract the exponents, don’t you? Guess what? When you divide expressions with

logarithms, you subtract the logarithms. And finally when, we have one more

property of logarithms since logarithms are exponents, when you exponent an

exponent, we end up multiplying them, don’t we? Well, when you exponent a logarithm you

end up multiplying them. Look at the other place where you can put the b and

this is going to be the most useful of all the properties because that b is

hard to get when it’s up in the exponent. Another place to put it is down in front

and we can solve an equation much easier. Get rid of the b, if you would, when it’s

down on the ground. Ok, so that’s another location we can put

it. Let’s use the properties of logarithms

to expand this problem. We’ll start with, we have an expression here with division.

Logarithm with division becomes a subtraction of the logs; division

becomes a…notice that it becomes an easier operation. This is good; this is a

good trend. Now, what else do we have in here? We have multiplication of X. Ooops, not

exponents multiplication of logarithms and of course it’s the same rule to, to take

the log of two numbers multiplied we’ll add the numbers. Now, what about the third rule? Where else can we put these exponents? The exponent rule tells us that we can

multiply. Once again, the next easier operation and actually this isn’t a rule

of logs. If you think about it, three log of a’s is log of a plus log of a plus

log of a, isn’t it? We can bring that one down. and minus 2 log of C’s is …so actually we

can take a very complicated calculation and turn it into a bunch of pluses and

minuses and this is going to be very important eventually because, because

that is the way your computer and your calculator work. It uses logarithms to

change these very difficult problems with exponents and division and

multiplication into a bunch of pluses and minuses or ons and offs and that is

a fact. An expansion like this may be interesting but for the most part we’re

going to be using logs, or the properties of logarithms to condense a problem.

We’re going to work backwards and contract these into one giant logarithm. Where can I put this two? Remember where else I’m allowed to put it. And what about the three? He can become an exponent and what does

addition become if we work backwards? Something the logs means multiply and basically if you think about it 3

squared x to the third is nine times eight. We’ve made this into one big fat

logarithm. Well, you’re going to want to do it in the future to solve equations,

believe me. This is called a contraction. Okay, let’s do another one, I work with the exponent first So a number out front can move to that

exponent spot and back and forth. Now, remember what we do with subtraction.

Subtraction becomes division and we’ve made this into one big fat

logarithm. It’s important for you to know that one log is going to be our goal. What does this contract to? Don’t let it

scare you. The log of c plus the log of a plus the log of b plus the log of i plus

the log of n. This is a classic mathematical problem. Remember what

addition becomes multiplication so this long expression actually becomes, well,

log cabin! Sorry about that. Okay, let’s do another. I want you to get good at contracting.

Make them into one big fat logarithm. Addition is going to become

multiplication 25 x 3 and subtraction is going to become division. We could write

it as a fraction and 75 divided by 5 is 15 so the answer here is the log of 15. Remember, we want to make it into one log eventually. You’ll see why when we go to do

equations. How about this one? Let’s see. Let’s work

on our exponents first. I say, you don’t see any exponents but there will be.

Where can I put that two? Yeah. Now I have another number out front.

Don’t let him scare ya. He goes up in the exponent position and

we know we can have half as an exponent. Now this one’s going to go up in the

exponent position for the whole expression inside parentheses. Now all

we’re going to have to do, oh, what, what is X to the one-half? Let’s

write that in simpler terms. X to the one-half is the square root of x, isn’t

it? So you could write it that way as well. Remember that’s a rule of exponents. Now

addition, now, (Quit crying!) addition becomes multiplication and subtraction becomes

division and you have changed this into one big

logarithm. Won’t get any harder than that! Okay, well I want you to go practice

these properties, especially the contractions into one logarithm because

we’ll be using that in the next show.

## 20 Comments

## Austin :D

That whole 'log cabin' one is it actually able to be condensed?

## McKenna Oliver

Logs finally make sense! Thanks Bill

## Tom

Informative–and hilarious. Thank you.

## TheJubiks

Thank you for making this video. It was easy to understand and the background comments/music was a nice touch ðŸ™‚

## Heidi Melendez

So clear and understandable, not to mention delightful. My sincere thanks.

## Siddhartha Ray

I have tried my best, sincerely

## torosalvajebcn

Excellent, thanks.

## Veda Phaninindra

it's excellent, thank-you for your good explanation, Witte.

## Dustin Watson

Nicely done video

## Pengu

I lost it at Log cabin

## Gee Money

Your delivery of your explanations are EXCELLENT

## Kit Wings

The examples are so funny! 0:56 – 1:12

## Mahroukh Khan

LOG CABIN I AM DYING. I didn't know it was possible for math to be funny, thank you!

## elsherif1969

Mr Bill is the best ever in YouTube … He was and still very useful to me … Thank you from the deep of my heart … I'm math teacher

## Jacket down Jerry

Dude breathing hard as hell

## YuÐ¼Î¹Ñ” áƒ¦

CÃ©line Dion – It's All Coming Back To Me Now lmao

## ANDY HO

THANKS

( Í¡Â°( Í¡Â° ÍœÊ–( Í¡Â° ÍœÊ– Í¡Â°)Ê– Í¡Â°) Í¡Â°)

## Bryan Zen

MS.CAI NOOOOOOOO!!!!!!!!!!!!!!!!!!!!!

## Kino -Imsure1200q

It's really obvious that logb(b^x) = x because b = b^1 and b^x = b^(1*x) meaning logb(b^x)=x

## Kino -Imsure1200q

in the last equation i got it to log(sqrt(x^5)/(x+1)^2)

## Anonymous boy

A very useful video…before I have never known that how to solve logirthm but now after watching this video, I understood the basic concepts of logirthm…and I could solve the sums with ease…thank u sir