Prove the existence of logarithms

In summary, logarithms are mathematical functions used to solve equations with exponential terms. They exist because they simplify complex calculations with large numbers. The existence of logarithms can be proven through mathematical proofs and examples. They have various real-life applications in finance, science, and engineering. There are different types of logarithms, such as natural, common, and binary, each with its own purpose and interchangeable through mathematical rules. Logarithms cannot be negative or imaginary in real-life applications, but complex logarithms can be defined for complex numbers.
  • #1
tronter
185
1
Fix [itex] b >1, \ y >0 [/itex], and prove that there is a unique real [itex] x [/itex] such that [itex] b^{x} = y [/itex].

Here is the outline:

(a) For any positive integer [itex] n [/itex], [itex] b^{n}-1 \geq n(b-1) [/itex]. Why do we do this?

(b) So [itex] b-1 > n(b^{1/n}-1) [/itex].

(c) If [itex] t>1 [/itex] and [itex] n > (b-1)/(t-1) [/itex] then [itex] b^{1/n} < t [/itex].

etc..


Is this the correct process?
 
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  • #2
my approach:
draw graph of b^x .. and use some theorm (I forgot it's name) .. I think intermediate value theorm
 

1. What are logarithms and why do they exist?

Logarithms are mathematical functions that are used to solve equations with exponential terms. They exist because they are a useful tool for simplifying complex calculations involving large numbers.

2. How can we prove the existence of logarithms?

The simplest way to prove the existence of logarithms is by using the definition of logarithms, which states that they are the inverse function of exponentials. This can be shown through mathematical proofs and examples.

3. What are some real-life applications of logarithms?

Logarithms are commonly used in finance, science, and engineering. For example, they are used in finance to calculate compound interest and in earthquakes to measure the intensity of seismic waves.

4. Are there different types of logarithms?

Yes, there are different types of logarithms such as natural logarithms (base e), common logarithms (base 10), and binary logarithms (base 2). Each type is used for different purposes and can be interchanged using certain mathematical rules.

5. Can logarithms be negative or imaginary?

Logarithms cannot have negative or imaginary values because they are the inverse function of exponentials, which are always positive. However, complex logarithms can be defined for complex numbers, but they are not commonly used in real-life applications.

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