Log to the power of two.

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MayaMaya121

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Hi, my native language isn't English so excuse me if I make grammar mistakes.

What does it mean when I have a logarithm to the power of two?
No, I don't mean that it looks like: log2 a.
It looks like: log 2 2 a. (The subscript is supposed to be exactly below the superscript)
 
MayaMaya121 said:
Hi, my native language isn't English so excuse me if I make grammar mistakes.

What does it mean when I have a logarithm to the power of two?
No, I don't mean that it looks like: log2 a.
It looks like: log 2 2 a. (The subscript is supposed to be exactly below the superscript)
Click to expand...
I would interpret it as follows:

log22a = [log2a]2 = [log2a] * [log2a]
 
MayaMaya121 said:
Hi, my native language isn't English so excuse me if I make grammar mistakes.

What does it mean when I have a logarithm to the power of two?
No, I don't mean that it looks like: log2 a.
It looks like: log 2 2 a. (The subscript is supposed to be exactly below the superscript)
Click to expand...
Please post complete text or screen capture of whatever you are referencing. And what exactly are you asking? If it's squared, then the meaning is that we calculate the log value, then we square it.
 
I can't post a photo of the question because I am now allowed to post question from the text book.
The part of the question I am interested in is solving the inequality: [MATH]4-\log _2^2\left(x^2\right)\ge 0[/MATH]And if I need to find the when [MATH]x^2>0[/MATH] from the same inequality, so is this specific inequality necessary even is the log is squared?

Thank you for your help.
 
MayaMaya121 said:
I can't post a photo of the question because I am now allowed to post question from the text book.
The part of the question I am interested in is solving the inequality: [MATH]4-\log _2^2\left(x^2\right)\ge 0[/MATH]And if I need to find the when [MATH]x^2>0[/MATH] from the same inequality, so is this specific inequality necessary even is the log is squared?
Thank you for your help.
Click to expand...
Find what?

Please read your post prior to posting for editing.
 
I didn't mean to write "the". I am sorry, I will read my replies more carefully before I post them.
 
\(\displaystyle log^2(x)\) mean the square of the logarithm of x- find the logarithm of x and multiply it by itself. There is no simplified formula for that, you just do the multiplication. For example, assuming "log" here means natural logarithm (which it commonly does in "higher math"- in secondary school math it often is the "common" logarithm) \(\displaystyle log^2(2)= (0.6931)^2= 0.4804\).
 
MayaMaya121 said:
I can't post a photo of the question because I am now allowed to post question from the text book.
The part of the question I am interested in is solving the inequality: [MATH]4-\log _2^2\left(x^2\right)\ge 0[/MATH]And if I need to find the when [MATH]x^2>0[/MATH] from the same inequality, so is this specific inequality necessary even is the log is squared?

Thank you for your help.
Click to expand...
It may help to use a substitution. If we let [MATH]u = \log_2(x^2)[/MATH], the inequality becomes [MATH]4 - u^2 \ge 0[/MATH]. Can you solve that? Then you can replace u with its meaning and continue to solve.

If you want to solve [MATH]x^2>0[/MATH] in order to find the domain of [MATH]4-\log _2^2\left(x^2\right)[/MATH], as part of the same problem, then, yes, that will need to be done at some point. What is done to the log does not change that fact that its argument must be positive.
 
Quick language lesson. In the US

[MATH]log_2(x)[/MATH] is a logarithm with a base of 2, frequently abbreviated to log base 2. See Hall's post for the difference between the common logarithm, base 10, and the natural logarithm, base e. A logarithm's base is specified by a subscript.

The definition of the logarithm with a base of 2 is

[MATH]log_2(x) = a \iff x = 2^a.[/MATH]
But at the end of the day, a logarithm is still nothing more than a number. You can square it.

Consider thinking about your problem like this.

[MATH]\text {Let } y = log_2(x) \implies log_2^2(x) = \{log_2(x)\}^2 = y^2.[/MATH]
Clear?

[MATH]\therefore 4 - log_2^2(x) \ge 0 \implies 4 - y^2 \ge 0 \implies 4 \ge y^2.[/MATH]
What do you get when you solve that?

Can you see how to proceed thereafter?

Dr. Peterson: I do wish that you would wait until after noon to look inside my mind. I need coffee, and lots of it, before it is respectable sight.
 
Thank you everyone for answering and helping me with my homework.
I solved the problem and got the correct answer.
 
JeffM said:
Dr. Peterson: I do wish that you would wait until after noon to look inside my mind. I need coffee, and lots of it, before it is respectable sight.
Click to expand...

I'll try at least to knock before entering.

In this problem, of course, we have both a base and a power of 2; I was initially wondering (from the title) if MayaMaya121 might have meant "base", but the question was very carefully written to anticipate that. (Thanks for that!)

One thing I've chosen not to mention until now is that at some levels, [MATH]log^2(x)[/MATH] can be taken to mean [MATH]log(log(x))[/MATH], that is, an iterated function. I am pretty sure that is not what is in view here, but it is one reason we tend to avoid that notation, and prefer to write [MATH](log(x))^2[/MATH].
 
MayaMaya121 said:
What does it mean when I have a logarithm to the power of two?
No, I don't mean that it looks like: log2 a.
It looks like: log 2 2 a. (The subscript is supposed to be exactly below the superscript)
Click to expand...
The code \log_2^2(x) produces \(\log_2^2(x) \)
 
I wanted only to understand what does log to the power of two mean so I wrote the part of the question that was related to the subject. Since I didn't need to post the whole question.
 
MayaMaya121 said:
I wanted only to understand what does log to the power of two mean so I wrote the part of the question that was related to the subject. Since I didn't need to post the whole question.
Click to expand...
Suppose that \(F\) is a function then \(F^2(x_0)\) means \([F(x_0)]^2\)
 
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