What is Log? Exploring Logarithms & Power Laws

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In summary, the conversation revolved around logs and logarithms, and the confusion and annoyance of the speaker when their teacher would casually mention them without fully explaining their purpose. The speaker also asked about the use of logs in engineering and received helpful resources for further understanding. The conversation also touched on the concept of natural logs and their significance in mathematics. Various tips and explanations were shared to better understand the concept of logs.
  • #1
Mozart
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A while back in the school-year we were doing logs, and logarithm. We learned the power law, and such. It really annoys me when my teacher just throws something on the board like log and then shows us laws to remember. I don't even know what log is. Does it stand for a number? or is it just something in between to allow you to switch things around. What is a natural log too? All I know is its base e which is 2.718. Last question is how do engineers use log in the real world to builds bridges, towers, etc.

Anyways ease on the flamming I am just trying to learn or better understand what I have learned. I feel like I understood it before but have forgotten since I never reviewed it after the test about 5 months ago.

Thanks.
 
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  • #3
Thank you!
 
  • #4
As you mentioned, a natural log is no different than any other log except for it's base must be 'e'. What makes 'e' so special? It answers this important question:

Suppose I have a function that looks like b^x.
For what value of b will d[b^x]/dx = b^x?
Or, for what value of b will the derivative of our function equal the function itself?

One can solve this equation for b and discover that b=~2.718. [*]
This property is so extremely useful that it pops up all the time. This is why it was given a dedicated constant.

* 2.718 is only an approximation, as I am sure you know. If you want to know what b, and therefore e, really is then follow this link http://www.answers.com/topic/e-mathematical-constant and be sure to click on irrational and transcendental once you are done reading the page.
 
  • #5
Mozart said:
Thank you!

My pleasure.
 
  • #6
The best way to think of logs is as follows.

Consider:

ln x = y

Most people would read this aloud as "natural log of x equals y".

But there is a better way to "sound it out". Try this -

"The power which you must raise e to obtain x is y". It helps.

After all, in this case, e^y = x, so you can easily remember what a log is by reading it aloud that way - what it really represents.
 
  • #7
It's funny you would say that. I found myself yesterday looking in the mirror repeatedly seeing that image in my head and saying it in my head. I don't really have a problem with understanding how to work out log problems. I just felt I didn't understand them in depth. This thread definitely helped me review for the final exam comming up in June. There will be about 2 log questions I think.
 
  • #8
My first college instructor liked to emphatically put it this way: [tex]e^{lnX}[/tex] says that e is the power to which ln(X) must be raised to give X!

Is that helpful?
 
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  • #9
robert Ihnot said:
My first college instructor liked to emphatically put it this way: [tex]e^{lnX}[/tex] says that e is the power to which ln(X) must be raised to give X!

Is that helpful?

No it's not, because it should read: lnx is the power to which e must be raised to give x!

:-p
 

Related to What is Log? Exploring Logarithms & Power Laws

What is a logarithm?

A logarithm is a mathematical function that represents the power to which a fixed number, called the base, must be raised to produce a given number. It is the opposite of the exponentiation function.

What is the purpose of using logarithms?

Logarithms are used to solve equations involving exponential functions, simplify complex calculations, and understand relationships between numbers that vary exponentially.

What is the difference between a common logarithm and a natural logarithm?

A common logarithm uses a base of 10, while a natural logarithm uses a base of the mathematical constant e (approximately 2.71828). They are denoted by log and ln, respectively.

How do logarithms relate to power laws?

Logarithms are closely related to power laws, as both involve exponential functions. In a power law, the dependent variable is proportional to a power of the independent variable. This relationship can be expressed using a logarithm, making it easier to analyze and interpret.

What are some real-world applications of logarithms?

Logarithms are used in a wide range of scientific fields, including physics, biology, economics, and engineering. They can be used to model population growth, measure the intensity of earthquakes, analyze financial data, and more.

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