Logarithmic Simplification: Understanding the Natural Logarithm

In summary, the conversation discusses the simplification of loga(aloga(x)) to loga(x). The concept of logarithms and their properties are explained, specifically the injective nature of the function. Two different approaches to proving why aloga(x) = x are discussed, one using the definition of inverse functions and the other using the laws of logarithms. A typo is pointed out and corrected.
  • #1
Bardagath
10
0
Hello,

I made a mistake in the title of this thread and this question is on general logarithms;

loga(aloga(x)) = loga(x) ==> aloga(x) = x

Can someone enlighten me on why loga(aloga(x)) simplifies to loga(x)? Can someone prove why this is true? Futhermore, why does this imply that aloga(x) = x? I am having trouble getting my head around this



Bardagath
 
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  • #2
Basically you are asking why

[tex]x = a^{\log_a(x)}[/tex]

What is log definition?
 
  • #3
Do you know the rules:
[tex]\log_b(a^x) = x\log_b(a)[/tex]
[tex]\log_b(b) = 1[/tex]
If you do, then it follows from first applying the first rules and then seeing:
[tex]\log_a(x) \log_a(a) = \log_a(x) \times 1 = \log_a(x)[/tex]

The logarithm is what we call an injective function (also called one-to-one function I believe) which basically means that if two elements a and b are mapped to the same element, i.e. [itex]\log_c(a) = \log_c(b)[/itex], then they must necessarily be the same (a=b) since no two different elements map to the same. Apply this to:
[tex]\log_a\left(a^{\log_a(x)}\right) = \log_a(x)[/tex]
if you have already shown that log is injective (otherwise you need to use some other property, but the argument seems to suggest that this is the property used).
 
  • #4
I guess that argument does this. If we know [itex]\log_a(a^x) = x[/itex] for all [itex]x[/itex], we want to use it to show [itex]a^{\log_a x} = x[/itex] for all [itex]x[/itex].
 
  • #5
g_edgar said:
I guess that argument does this. If we know [itex]\log_a(a^x) = x[/itex] for all [itex]x[/itex], we want to use it to show [itex]a^{\log_a x} = x[/itex] for all [itex]x[/itex].
There are two ways of approaching this. One is that [itex]log_a(x)[/itex] is defined as the inverse function to [itex]a^x[/itex]. By the definition of "inverse functions", which requires that f(f-1(x))= x and that f-1(f(x))= x, then, [itex]a^{log_a(x)}= x[/itex] and [itex]log_a(a^x)= x[/itex].

Or, just using the laws of logarithms (which are, after all, derived from the definitions), [itex]log_a(a^x)= x log_a(a)= x[/itex] and, if [itex]y= a^{log_a(x)}[/itex], taking the [itex]log_a[/itex] of both sides, [itex]log_a(y)= log_a(a^{log_a(x)})[/itex][itex]= (log_a(x))(log_a(a))= log_a(x)[/itex] and, since [itex]log_a[/itex] is "one-to-one" function, y= x.
 
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  • #6
Obvious typo - you meant a^, not e^ :)
 
  • #7
I don't know why it didn't fall into place earlier but I woke up today and it fits;

Yes, loga(x)loga(a) = loga(x) . 1 = loga(x)

Thankyou very much for your replies!
 
  • #8
Borek said:
Obvious typo - you meant a^, not e^ :)
Yes, of course. Thanks. I have edited my post so I can pretend I didn't make that mistake.
 
  • #9
Now your LaTeX is broken :rofl:
 

1. What is a natural logarithm?

The natural logarithm is a mathematical function that is the inverse of the exponential function. It is denoted as ln(x) and is the logarithm with base e, where e is an irrational number approximately equal to 2.71828. It is commonly used in many fields of science and mathematics.

2. What is the difference between natural logarithm and common logarithm?

The main difference between natural logarithm (ln) and common logarithm (log) is the base. Natural logarithm has a base of e, while common logarithm has a base of 10. This means that ln(x) represents the power to which e must be raised to equal x, while log(x) represents the power to which 10 must be raised to equal x.

3. How is natural logarithm used in real life?

Natural logarithm is used in many real-life applications, such as in finance, biology, and physics. In finance, it is used to calculate compound interest and continuous growth. In biology, it is used to model population growth and decay. In physics, it is used to model radioactive decay and other exponential processes.

4. What is the derivative of the natural logarithm function?

The derivative of the natural logarithm function ln(x) is 1/x. This means that the slope of the tangent line at any point on the ln(x) curve is equal to 1/x. This is a useful property in calculus and is used to solve many mathematical problems involving exponential and logarithmic functions.

5. How do you solve equations involving natural logarithms?

To solve equations involving natural logarithms, we use the properties of logarithms, such as the power rule and the product rule. We also use algebraic manipulation to isolate the logarithm and then use the definition of a logarithm to solve for the variable. It is important to be familiar with the rules and properties of logarithms to successfully solve these types of equations.

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