what does ln^2(x) mean?

Question

does it mean (ln x)^2 or ln(x^2) or something totally different?

Answer 1

Before I answer, I want to note that if the logarithm has a SUBSCRIPT, it typically means that the logarithm has the subscript as a base. For example, natural log (usually written ln) is a logarithm with base e, so it could be written log_e, where the _ denotes a subscript.

In your case, with a superscript, it depends on who is writing it. Most likely,

ln^2 x = (ln x)^2 = (ln x)*(ln x)

This notation (a superscript on the function) became popular to use in trigonometry with sin(x) and cos(x) functions, where sin^2(x)=sin(x)*sin(x). Since then, people often use it with other functions.

However, more formal mathematicians typically say that (with an exception made for the trig functions),

ln^2(x) = ln( ln( x ) )

In other words, ln^2 means that the operation "ln" was applied to "x" twice. For example, remember that in linear algebra, if you have an n-by-n matrix A and an n-by-1 vector x, the multiplication:

A*x

Since this results in another n-by-1 vector A*x, it can be thought of as the matrix A "operating" on x, and sometimes it is written:

A(x)

However, since A(x) (or A*x) represents another n-by-1 vector, A can also operate on A(x) (or A*x). If A operates on A*x, then it could be written as:

A*A*x = A^2 x

Now, you might want to write the thing on the left as A(A(x)). However, then that means you'd want to write the thing on the right as A^2(x). Thus, A(A(x))=A^2(x). Hence, the superscript notation does not refer to "squaring" the A(x). Instead, it refers to A operating on x "twice."

If you're in an upper-level math class, you will probably use the latter definition (which many mathematicians would argue is "more correct"). Otherwise, if you're an average non-mathematician, you will probably use the former definition.

Answer 2

Definition Of Ln X

Answer 3

It means a typo or bad notation.

Maybe you mean log (base 2) of x?
The way that it is written does not make sense since you cannot raise ln to a power (ln is the name of the function).

Answer 4

it means
(ln x) (ln x)

ln(x^2) is totally different from (ln x) (ln x) = ln^2 (x)

Answer 5

it doesn't mean anything

ln function requires an arguement

In this case ln is raised to 2 (x) which means no arguement for ln

Answer 6

ln(x) * ln(x) I suppose ...

Answer 7

SPEAK ENGLISH PLZ!