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"
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?
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 .
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:
Let's concentrate on the 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 . Then So, .
Let . Then . So, again
If you say , then will again equal 1.
So, we ALWAYS have . As you see, the 1 on the right side of the equaltion is our x in the previous formula ( ), 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 .
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:
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.