The way we write ln(x) to the power of a

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In summary, while (cos x)a, (sin x)a, and (tan x)a can be written as cosax, sinax, and tanax, respectively, we do not typically write (ln x)a as lnax or (log x)a as logax. This convention is commonly used in textbooks and can make the final writeup cleaner. However, the notation fa(x) can be ambiguous and it is important to consider the context when interpreting it.
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I have learned that (cos x)a can be written as cosax. And like wise for sin x and tan x.

How come we don't write (ln x)a as lnax or (log x)a as logax?

I just think it seems smarter when its written with as few symbols as possible.

(Or perhaps its just my textbook which don't write it this way?)

Thanks in advance!
 
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  • #2
LOTS of textbooks use that convention. I will go as far as to say it is the most common convention. :smile:
 
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  • #3
Personally, I hardly ever see logs with powers (except in a couple of circumstances), while trigs with powers (especially squared) are a common occurrence.
 
  • #4
My advisor has gone so far as to tell me when I square any function I should write it as f2(x), which I personally don't like but what am I going to do about it (I would rather write f(x)2). I will admit that it makes the final writeup a little cleaner looking
 
  • #5
The notation fa(x) is a little bit ambiguous because it can mean either f(x)a or f(f(...f(x)...)).

You can usually work out the meaning from context. If f is a trig function, you are probably okay, since in most situations it does not make sense to iterate the sine function. But both log(log(x)) and (log(x))2 are potentially useful things.
 

1. What does "ln(x) to the power of a" mean?

"ln(x) to the power of a" means taking the natural logarithm of x and raising it to the power of a. This is equivalent to e^(a*ln(x)), where e is the natural logarithm base.

2. Why do we use ln(x) to express exponents?

Using ln(x) to express exponents allows us to simplify and solve equations more easily, especially in calculus and mathematical analysis. It also has applications in fields such as physics and economics.

3. Can we use any other base for this expression?

Yes, we can use any base for this expression. However, using ln(x) (base e) is preferred because it has unique properties that make it useful in various mathematical operations and applications.

4. What is the difference between ln(x) to the power of a and e to the power of a?

The main difference is that ln(x) to the power of a takes the natural logarithm of x before raising it to the power of a, while e to the power of a simply raises e (the natural logarithm base) to the power of a. This results in different numerical values for the same exponent a.

5. How is ln(x) to the power of a used in real-life situations?

ln(x) to the power of a has various applications in fields such as finance, biology, and physics. For example, in finance, it can be used to calculate continuously compounded interest rates. In biology, it is used in the logistic growth model to describe population growth. In physics, it is used in the decay of radioactive substances.

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