How can log(x) = ln(x)/ln(10)?

  • Context: High School 
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Discussion Overview

The discussion revolves around the relationship between logarithmic functions, specifically the equation log(x) = ln(x)/ln(10). Participants explore ways to understand and prove this relationship without relying on numerical examples.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • Some participants inquire about proving the equation log(x) = ln(x)/ln(10) without using specific numbers.
  • One participant presents the equation x = 10^(log x) as a definition of logarithm, suggesting it may help in understanding the relationship.
  • Another participant references a known formula for changing the base of logarithms, log_a(b) = log_c(b)/log_c(a), as a potential method for proving the relationship.
  • There is a clarification regarding the direction of base change in the logarithmic formula, with participants discussing the correct interpretation.

Areas of Agreement / Disagreement

Participants generally agree on the validity of the change of base formula, but there is no consensus on the best approach to prove the specific relationship in question without numerical examples.

Contextual Notes

Some assumptions about the definitions of logarithms and bases may not be explicitly stated, and the discussion does not resolve the proof of the relationship.

Natalinatul
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Might be a silly question but I'm trying to search for ways to prove that it's right without using the numbers itself
 
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Natalinatul said:
Might be a silly question but I'm trying to search for ways to prove that it's right without using the numbers itself
Does the following equation make sense to you:
$$x=10^{log x}$$
It says that, by definition, the log of x is the power to which you have to raise 10 to get x.
 
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Chestermiller's answer is correct, and so by this time you may have figured this out, but if you are still having difficulty, just google the proof of the change of base formula. Or just google the change of base formula, and many sites will also offer the proof (based on Chestermiller's hint)
 
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Yup.. How could I miss that! The best thing about his answer was the fact he didn't answer it but gave a great hint... I figured it out now...
 
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Hi, there is this formula ##\log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}## that permits to change the base from ##b## to ##c##.
 
Ssnow said:
Hi, there is this formula ##\log_{a}{b}=\frac{\log_{c}{b}}{\log_{c}{a}}## that permits to change the base from ##b## to ##c##.
Don't you mean when you change the base from c to a?
 
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