Thread: Logarithms

1. Logarithms

Hi, I don't really understand what the textbook means by this quote, could someone explain it to me and give me an example?

"The logarithm of x to the base 1 is only valid with x=1, in which case y has an infinite number of solutions and is not a function"

2. Originally Posted by skeske1234
Hi, I don't really understand what the textbook means by this quote, could someone explain it to me and give me an example?
"The logarithm of x to the base 1 is only valid with x=1, in which case y has an infinite number of solutions and is not a function"
We must have the question to which that is the answer.
Please do not post an answer and expect us to know the question.
Your textbook is clearly explaining an answer to some question.
What question?

3. Originally Posted by Plato
We must have the question to which that is the answer.
Please do not post an answer and expect us to know the question.
Your textbook is clearly explaining an answer to some question.
What question?
It's part of the "key concepts" area of the end of the chapter...
It just has a bullet point and says:

- Exponential and log functions are defined only for positive values of the base that are not equal to one:
y=b^x, b>0, x>0, b cannot equal 1
y=log_bx, b>0, y>0, b cannot equal 1

The logarithm of x to the base 1 is only valid when x=1, in which case y has an infinite number of solutions and is not a function.

I just don't get the last part, what is it talking about when it says that?

4. From the basic definition $\displaystyle b > 0\; \Rightarrow \;\log _b (x) = y\;\iff\;b^y = x$.
That means that $\displaystyle \log _1 (x) = y\;\iff\;1^y = x$, but that means $\displaystyle x=1$ for any $\displaystyle y$.
Does that help?

5. Hi,

I will use the answer of Plato and extend it a little bit. My answer is a little bit longer, but I hope you'll read it till the end .

Originally Posted by Plato
From the basic definition $\displaystyle b > 0\; \Rightarrow \;\log _b (x) = y\;\iff\;b^y = x$.
That means that $\displaystyle \log _1 (x) = y\;\iff\;1^y = x$, but that means $\displaystyle x=1$ for any $\displaystyle y$.
We have a restriction that the base b cannot equal 1. Let's ignore this restriction, and see what will happen, if the base b equals 1. As Plato wrote:

$\displaystyle \log _1 (x) = y\;\iff\;1^y = x$

Let's concentrate on the $\displaystyle 1^y = x$ part. No matter what value you assign to y, the result will always be 1, because no matter how many times you multiply 1, the result is always 1.
For example, let $\displaystyle y = 3$. Then $\displaystyle 1^3 = 1.1.1 = 1.$ So, $\displaystyle x = 1$.
Let $\displaystyle y = 5$. Then $\displaystyle 1^5 = 1.1.1.1.1 = 1$. So, again $\displaystyle x = 1.$
If you say $\displaystyle y = 8$, then $\displaystyle 1^8$ will again equal 1.
So, we ALWAYS have $\displaystyle 1^y = 1$. As you see, the 1 on the right side of the equaltion is our x in the previous formula ($\displaystyle 1^y = x$), so x is always 1.

So, let's return to the unclear statement from your book. I think the first part should be clear now:

"The logarithm of x to the base 1 is only valid with x=1, ..... ".

We have just seen that x is always 1 in each case (no matter what we assign to y). That's why they say that the logarithm of x to the base 1 is ONLY valid when $\displaystyle x = 1$.

Lets' continue with the statement:
"The logarithm of x to the base 1 is only valid with x=1, in which case y has an infinite number of solutions ..... ".

We have seen that we can assign any number to y: y = 3, y = 5, y = 8, ... etc., and we get always the number 1 as an answer, which is the correct answer, because no matter how many times we multiply 1, we receive 1. So, y really has an infinite number of solutions.

Let's continue with the statement:

"The logarithm of x to the base 1 is only valid with x=1, in which case y has an infinite number of solutions and is not a function".

Here you have to remember the definition of a function. I will cite Wikipedia (http://en.wikipedia.org/wiki/Function_(mathematics)):

"A function is defined as a relation between two terms, which are called variables because their values vary. Call the terms, for example, x and y. If every value of x is associated with exactly one value of y, then y is said to be a function of x."

Concentrate on the last sentence. You cannot have for the same value of x, in our case 1, more than one value for y. But as we have seen, you have lots of solutions for y, when x is 1! Not just one solution. You can have y = 3, y = 5, y = 8, ... etc. So, if we allow the base b to be equal to 1, these expressions will not express functions, in the case when the base is 1:

$\displaystyle \log _1 (x) = y$

$\displaystyle 1^y = x$

The two variables x and y are still in a relation, but one of them is not a function of the other. And you see - in order to define the logarithmic and exponential functions, you need the restriction that the base is not 1. Otherwise, we cannot call them "functions". This is the last part of the statement from the book: they say that y is not a function of x, because y has an infinite number of solutions.

I hope this helps, please, ask if something is unclear.