Prove that the logarithmic function is continuous on R.

In summary, the author is trying to find a value for a \delta>0 such that -\delta<x-x_0<\delta. However, this is not possible because there is no way to generate a value for \left|\log_{a}x-\log_{a}x_{0}\right| < \varepsilon.
  • #1
Portuga
56
6

Homework Statement



Prove that [itex]f\left(x\right)=\log_{a}x[/itex] is continuous for all [itex]\mathbb{R}[/itex].

Homework Equations

[/B]

I must find a [itex]\delta>0\in\mathbb{R}[/itex] for a given [itex]\varepsilon>0[/itex]
such that
[tex]
\left|x-x_{0}\right|<\delta\Rightarrow\left|\log_{a}x-\log_{a}x_{0}\right|<\varepsilon.
[/tex]

The Attempt at a Solution

.[/B]

I tried to use a direct proof solving [itex]\left|\log_{a}x-\log_{a}x_{0}\right|<\varepsilon[/itex] for [itex]x[/itex]. But this gives rise to a pair of values: [itex]\delta=x_{0}\left(a^{\varepsilon}-1\right)[/itex] and [itex]\delta=x_{0}\left(a^{-\varepsilon}-1\right)[/itex]. When I use them to build [itex]\left|x-x_{0}\right| < \delta[/itex] from the inequality
[tex]
-\delta<x-x_{0}<\delta
[/tex]
I see myself in big trouble, as there is no way to generate [itex]\left|\log_{a}x-\log_{a}x_{0}\right| < \varepsilon[/itex].

Can someone give me a hint, or a new strategy? I have searched for help in other forums, but hints are very sophisticated to follow.
Thanks in advance.
 
Physics news on Phys.org
  • #2
How do you expect a function that is not defined at ##x =0## to be continuous on the entire real line?
 
  • Like
Likes Portuga
  • #3
Orodruin said:
How do you expect a function that is not defined at ##x =0## to be continuous on the entire real line?
Oh, sorry, I forgot it! Continuous for all [itex] x>0 [/itex].
 
  • #4
Why don't you try and see where ##\log x - \log x_0 = \log \frac{x}{x_0}## gets you? Those proofs can often be done by writing it the wrong way and start with ##|f(x)-f(x_0)| < \varepsilon## to obtain a condition for ##\delta## to get an idea and then turn the estimations, resp. conclusions around.
 
  • #5
You have solved [tex]
-\epsilon < \log_a (x/x_0) < \epsilon[/tex] correctly to get [tex]
x_0(a^{-\epsilon} - 1) < x - x_0 < x_0(a^{\epsilon} - 1),[/tex] but the interval is not symmetric about zero.

In this situation the closest end point to zero gives you your [itex]\delta[/itex]. For suppose [itex]-b < x - x_0 < c[/itex] for strictly positive [itex]b[/itex] and [itex]c[/itex]. If [itex]b < c[/itex] then [itex]|x - x_0| < b[/itex] implies [itex]-b < x - x_0 < b < c[/itex] as required; alternatively if [itex]b > c[/itex] then [itex]|x - x_0| < c[/itex] implies [itex]-b < -c < x - x_0 < c[/itex], again as required.
 
  • #6
Thank you very much, pasmith! I was so close to the answer! Now I understand. Thank you very much.
 

1. How is continuity defined for a function?

Continuity is defined as the property of a function where the output values change continuously as the input values change. In other words, there are no sudden jumps or breaks in the graph of the function.

2. What is the definition of a logarithmic function?

A logarithmic function is a mathematical function that represents the inverse of an exponential function. It is defined as f(x) = logb(x), where b is the base of the logarithm and x is the input value.

3. How do you prove that a logarithmic function is continuous on R?

To prove that a logarithmic function is continuous on R, we need to show that the limit of the function as x approaches any real number a is equal to the value of the function at a. This can be done by using the properties of logarithms and the definition of continuity.

4. What are the properties of logarithmic functions?

The main properties of logarithmic functions are:

  • The logarithm of a product is equal to the sum of the logarithms of each factor.
  • The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
  • The logarithm of a power is equal to the product of the exponent and the logarithm of the base.

5. Why is it important to prove that a logarithmic function is continuous on R?

Proving that a logarithmic function is continuous on R is important because it allows us to use the properties of continuity to make mathematical operations and solve equations involving logarithms. It also helps us understand the behavior of logarithmic functions and their graphs.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
5
Views
488
  • Calculus and Beyond Homework Help
2
Replies
58
Views
3K
  • Calculus and Beyond Homework Help
Replies
13
Views
961
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
222
  • Calculus and Beyond Homework Help
Replies
3
Views
5K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
551
  • Calculus and Beyond Homework Help
Replies
4
Views
878
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Back
Top