Why Not Zero?
This is a logarithmic equation:
In the comments to this post, everyone must reply, answer the question:
Why can "a" never equal zero?
An interactive learning ecology for students and parents in my Pre-Cal 40S class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.
This is a logarithmic equation:
im not really sure why "a" can't be zero but im going to give it a shot anyway!
ReplyDeletewe know that logb^a = c means the same thing as B^c = a.
so lets say that logb^a = c is log2^0 = 1 this is the same thing as 2^1 = 0! WRONG!! 2^1 is 2!!!!
2^1 = 2 can also mean log2^2=1!!!!
that is why "a" can not equal zero because it does not follow the universal law of math :D
thats what i think, im not sure if its right ^_^
bbye!!
Of my understanding, the reason Why can "a" never equal zero?, is because in a logarithm the argument can never equal zero because it is the output of the exponential form, b^c=a, if we look back no base to a power can equal zero, though it can come relatively close it will never touch equal zero, it can even be seen in a graph.
ReplyDeleteFor example:
10^1 =10
10^-1=1/10 10^0=1
So yeah what im trying to say is no matter what value you input for any number as its base it will never give you an ouput of zero in exponential form. To convert back to logarithm form, the output is the argument, therefore the argument in a logarithm can never equal zero.
Zero and negative numbers don't have valid logarithms!
Let's go back to the definition of a logarithm. It's the POWER that you raise a base to, in order to get the number you want.
ReplyDelete10^3 = 1000, so log of 1000 = 3. (In this case, we are using the base 10)
If we try to reduce the power, this is what we will get.
10^2 = 100
10^1 = 10
10^0 = 1
10^-1 = 0.1
10^-2 = 0.01
As for the log of zero, that would be the power that you have to raise 10 to in order to get zero.
But there's no way to raise a number to any power and end up with zero. So log(0) is undefined.
i agree with jhoann, the reason why "a" can never equal zero is because any base raised to any number no matter what will not give you a power of zero. Even if you stick in the smallest number known to man kind, you'll never get zero as a power. I liked how ruschev explained it using his example, if log(base2)0 = 1, then in exponential form it should be 2^1 = 0. but that's incorrect because 2^1 is equal to 2. and the logorithmic form of 2^1/= 2 is log(base2)2 = 1.
ReplyDeleteUm..
ReplyDeletelogb^a=c is the same thing as b^c=a
and from that. you would get log2^0=1 and changing that to exponential form, you'll get 2^1=0.. BUUT, that can't equal to zero, because 2^1 equals TWO.
Let's try other numbers..
2^2=4
2^0=1
2^-1=1/2
2^-2=1/4
whatever number you put (either positive or negative) into the exponent. you can't make it equal to zero.
i don't know, i'm not sure =/ i gave it a shot anyways..
ok, my turn! i was kind of confused at first, and im pretty sure everyone else was too. we might have known the answer but just afraid to say it. i guess i was scared to be wrong :P but anyways pretty sure everyone else was either unsure or just scared to say also.
ReplyDeletefrom my reading and understanding, everyone else seems to have the same idea, i thought so too because of the discussion we had in class.
whatever base you use, whether its 10, 2, 4 whatever you will never be able to touch 0. with this graph
http://upload.wikimedia.org/wikipedia/commons/e/e0/Logarithms.png
you can see that the log function will never touch the y-axis.
and back to everyone elses examples. a base multiplied by a number will never equal 0.
ex. base 10
10^-2=0.01
10^-1=0.1
10^0=1
10^1=10
10^2=100
and it works for all other bases. it just never touches 0. so "a" cannot = 0 or else the log would be incorrect.
hope this all made sense :P
Yeah I agree with the others, since the equation shown basically mean b^c=a then it is impossible for a to be 0 as there exists no number that when used as an exponent will get the number 0 as an answer.
ReplyDeleteFrom what I see 0 is pretty much an asymptote, numbers will get extremely close to 0 however none of them is ever going to touch it.
Okay,I don't really know the reason why "a" cannot be zero,i think its beacuse no number to the power of something will never equal 0, i think its because after zero there really isnt a whole number, not even a nigative number or fraction that would result in a zero not even 0^0.I think that if you made a number to the power of -.0000000000000etc..1 the result would not be zero because there are an infinite amout of decimal places. So in other words 0 is the only fixed number then the others would be -infinity or +infinity so the number really doesnt exist, because its ithere 0, positive or negative number nothing else. Also a fraction would still be ither before or after 0, and we know that any number to the power of 0 will equal one. I dont know if anyone can follow what im saying heh, but i tried. please tell me if i wrong....please.
ReplyDeleteThe "a" can never be equal to zero because any number you input in the exponent the result will be any number more than zero or any number less than zero. It is because number goes infinitely. It will not stop at zero then continues again to positive or negative infinity. Another is that logb (0) is undefined because there is no number X such that b^x= 0.
ReplyDeleteYou can visit this website to check for the source in which I came up with this answer:
http://en.wikipedia.org/wiki/Logarithmic_identities
In this website, it states that
logb (1)= 0 because b^0=1
logb (b)= 1 because b^1=b
If these are true, therefore, any number we input in the exponent will never come up with zero as the value of the argument.
The "a" can never be equal to zero because any number you input in the exponent the result will be any number more than zero or any number less than zero. It is because number goes infinitely. It will not stop at zero then continues again to positive or negative infinity. Another is that logb (0) is undefined because there is no number X such that b^x= 0.
ReplyDeleteYou can visit this website to check for the source in which I came up with this answer:
http://en.wikipedia.org/wiki/Logarithmic_identities
In this website, it states that
logb (1)= 0 because b^0=1
logb (b)= 1 because b^1=b
If these are true, therefore, any number we input in the exponent will never come up with zero as the value of the argument.
Okay, man i think i can explain it more clearly heh. We know that 0 is the MIDDLE of the number system right? so this is how it would look like:
ReplyDelete5,4,3,2,1,0,-1,-2,-3,-4,-5
(and fractions in between the numbers)
If 0 is the MIDDLE, and that x^0=x
then there isn't any other number that result in 0 because once we go to negative values the number will continue to shrink but never acutually become nothing(0=nothing)
because -1000000000000000 is and extreamly small number, but it will continue to shrink!!!!!!!!!so negative 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
is still a number,( buy now you should have cought on) so as you can see there really isnt a number that you can use as a power that will result to 0. Im 100% sure about this one heh.
I read everyone's comments and I agree with them. logb^a may come close to zero but it never reaches it, like an asymptote (an imaginary line where a graph comes significantly close but never touches it).
ReplyDeleteYou can have an exponent be 0 but that will always make the number equal 1. Using negative numbers will just make a fraction. So I don't think there is any base or exponent that will make 'a' equal 0.
oh and i think thats how an asymptote works, but and asymptote can be minipulated so that it dosnt just occure on the x axis and y axis, like the tan graph.
ReplyDeleteokay, this is my last little part of information.
Ok, here's how i see it... in the equation log b^a=c "a" can never equal zero, as no imput, no matter how big or small, can have an output of 0. I really like how allen explained that 0 was the middle of the number system, and that to the left (negative) the values get continually smaller below 0, and on the right (positive) the numbers get continually larger above 0, and that no number with an exponent, can reach that middle ground. Like the asymptote that is created on the x axis with a log function, an imaginary wall is created on both sides of the 0 enclosing it in this middle space. No number with an exponent will ever equal zero, or be found in this enclosed space...
ReplyDeleteok well it sounds a little weird but i did my best... good luck everyone, good night!! :D
Everyones comment is almost similiar to the answer of Ruschev because no matter what number you plug into the value of a and your exponent b will never equal the power 0 in exponential form of the logarithm(the easiest way to solving a log). So thats what I think I Cannot think of any other explanation So I gave it my best... Also because I'm doing someones scribe... ahem(tenn).
ReplyDeleteFrom my understanding, to answer the question I think that "a" can never equal zero because in order to have "a", the power equal to zero, the exponent "b" must equal to zero.
ReplyDeleteFor example:
b^c = a
(0)^1 = 0
And so if my reasoning is right...if the logarithm is of base zero to the power of zero they will cancel therefore only leaving the exponent. This can't really exist because how can an exponent exist if the logarithm does not?
Also if we do always have a power of zero and a base of zero but apply different exponents, it doesn't really make any sense because the output would always equal to zero no matter what input you plug in.
If my thoughts on this is completely wrong which it probably is then I guess I don't really know the answer to this question. Maybe Mr. K is expecting us to contradict the fact that "a" can equal to zero. I have no idea but as for everyone else's answer I don't agree but I don't disagree because I feel that they haven't 100 percent made me agree with the answer yet it does not look completely wrong as well.
I think I am "flummoxed" by this question =S
OKay, well I believe that you can never get the answer 0 because whatever number for the exponent you substitute it with it will never equal 0 .. not even 0, as an exponent will give you the power 0, it'll give you 1. I hope everyone gets my "drift" lol. Oh and everyone did great examples and explanations !! =)
ReplyDeleteThis comment has been removed by the author.
ReplyDeleteI know it to be that the argument in a logarithm cannot equal zero EVER, because there just is no value that you can plug into a logarithm that will actually do so. As a lot of people have given examples, I won't bother to emphasize how you go from positive intergers to insignificantly small fractions sticking in postive and negative values into b and c. Using all real numbers will never get zero as the arugent.
ReplyDeletewhy "a" can't be zero people say cause if put an number to the power of zero(ex. 2^0), the exponent will be one, same goes if your base is 4 or 10 or 100 etc. to the power of zero it still be the same exponent of one.
ReplyDeleteb^a=c is the some as clogb^a
but if you do both ways(equations)with the same variables the output be different by a long shot.
ex. 2^0=1 (b^a=c)and log2^1=0 (logb^a=c)
its the same when changing logarithmic into exponential but the answers will be different.
so my final saying to why "a" can't be zero is that any base to the power of zero will always give the exponent of one ALWAYS, AlWaYs, always... but i give a shot on answering the question ;P hahaha!!
seeeyaa!!!!
I thought this question will be easy so I decided to do it later, but now It's troubling me...
ReplyDeleteI'm still not satisfy with my answer..and how asymptote is related in this question.
I review some of my notes and this is my conclusion..
let's start it this way...
division is the inverse of multiplication..
If we can do this
2x3 = 6
then we can do this also
6/3 = 2, 6/2 =3
If we can do this
3x0 = 0
we can do this
0/3 = 0
but we cannot do this
0/0 = 3
because, if we do that.. then these are also true:
4x0 = 0
0/4 = 0 , 0/0 = 4
therefore, any number divided by zero can be any number. The answer is infinity and undefined.
Now let's try to use this principle to the exponents and logarithms..
Logarithm function is the inverse of exponential function.
Let's say
2^4 = 16
the invesre is
log(base2)16= 4
Now let's put zero into consideration..
0^4 = 0 (this is true)
log(base0)0 =4 (this is not true)
if we raise a O base to any exponent the answer will be zero. Therefore, if we turn this equation to log equation, the exponent could be any number. So the answer is infinite and undefine.
Now this "infite and undefine" that we're talking about is the asymptote. Asymptote is a line where a graph goes to a number near infinity though cannot be reach. This number is too large to compute that's why it is undefine.
whhoooh! that's all i can say..I can't explain it any further.
the reason why "a" can never equal is zero from my understanding is because it just can't. Just cuz guys! To be more ellaborate like many of you said if you plug any number to the input like let's say, a positive number or a negative number or even a fraction the output will never be zero. i guess for example 1/4^2=1/8 or 2^-2=1/4. i know those are bad examples but hey it's late. besides skimming through what you've all said the basic idea is that any number plugged into the input will never equal zero because there are infinite numbers and an exponent by definition which i copied from dictionary.com is "a mathematical notation indicating the number of times a quantity is multiplied by itself". with that said the input will always be multiplying itself over and over again by the exponent given and when you multiply you're increasing the number or decreasing it but it will never equal zero. so that's my rendition.
ReplyDeletei hope it makes sense and it's late so i'm off to bed. good night guys!
I think i know why 'a' cannot be 0.
ReplyDeleteAs many have said it is sort of an asymptote. An asymptote is an invisible line that in no way can a value be plugged into a variable be to solve a question that would equal 0. Whether it is unreachable, or it would become undefined.
This is true for this problem. It cannot be 0 because of these proofs:
let us give those variables numbers suppose it is log(base2)0= X
now what exponent would 2 need to be to be an output of 0? We'll go down, 2^2 would = 4, ^1 would equal 2, ^0 would equal 1. My gosh, we're out of numbers!!! But wait theres the fractional numbers, lets try those! BUt oh no! they become radicals!, No matter how little a fraction we put in as the exponent we'll only get a small positive number! It'll never reach 0 even if u go on infinitly.
And if u decide to put it in as a negitive decimal, all it does is give the same infinite result. Therefore, vitrually there will never be an answer to a logarithm equalling 0.
Well today i had to drive my sister to the doctors, thats why i didn't make it to class. Well good thing i saw this post. I read through people's comment and I agree. It's true that (A) can't be 0. In order it to be 0 ,the power has to equal zero,and the exponent which is (B) must equal to zero as well. Reason is that if it does not link together , then it wouldn't be the right answer.
ReplyDeleteEX: its like saying 2 x 3 = 6, but if the 3 wasn't there ,and if it was another number like
EX: 2 x 5 = 6 , it just wouldn't work without the 3.
well gotta get back studying by3
ps: sorry will
the reason why i think "a" cant be equal to zero, is because there is no base to any powers can be equal to zero.
ReplyDeletebase on the law of logarithm, logb^a=c is the same thing as b^c=a, therefore, log 5^0=1 is the same as 5^1=0, but this cant be right since 5^1 is 5, any base with the power of 1 is equal to itself!..
also, we know that 5^0 = 1, and 5^1=5, and as we keep increase the power, the result never can be equal to zero!
this is how i come with why "a" can't be equal to zero, but there's still things that got me confused as i keep trying to discuss into it.. anyways hopefully, we'll get a covinced answer by tomorrow class!
goodnight everyone :)
actually, ireally dont know what the reason is.. for me, b is the base and a is the number..so thats why a can never be equal 2 zero.. i need help guys..thanks
ReplyDeleteI'm not really sure of my answer anymore because i just woke up..but i think "a" can not be zero because a base can not be raised to a power that will give you an output of zero. i agree with jhoann's formula b^c=a and i think that "c" can not equal to anything to make "a" = 0
ReplyDelete