Proving y=ln|x| and its Limits: A Comprehensive Guide

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  • Thread starter Thread starter Ayinajin
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SUMMARY

The discussion focuses on proving the mathematical properties of the function y = ln|x|, specifically its derivative and limits. It establishes that the derivative y' = 1/x is derived using the chain rule, where f(x) = exp(ln(x)). Additionally, it confirms that lim ln|x| approaches +infinity as x approaches positive infinity, and lim ln|x| approaches -infinity as x approaches 0 from the positive side. The conversation emphasizes the importance of understanding these concepts for further mathematical exploration.

PREREQUISITES
  • Understanding of logarithmic functions, specifically natural logarithms.
  • Familiarity with calculus concepts, including derivatives and limits.
  • Knowledge of the chain rule in differentiation.
  • Basic comprehension of exponential functions, particularly exp(x) = e^x.
NEXT STEPS
  • Study the properties of natural logarithms and their derivatives.
  • Learn about the application of the chain rule in calculus.
  • Explore the concept of limits in calculus, focusing on logarithmic functions.
  • Investigate the relationship between exponential and logarithmic functions in depth.
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus, as well as anyone interested in the properties of logarithmic functions and their applications in higher-level mathematics.

Ayinajin
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I’ll appreciated very much if anyone can help me with any of these following proves by any means. Thank you!

prove:
1. y=ln|x| = y'=1/x

2. lim ln|x|=+ infinity as x approaches toward positive infinity and-
lim ln|x| = - infinity as x approaches 0 from positive side
 
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Maybe something like this... Let f(x) = exp(ln(x)), where exp(x) = e^x. Using the chain rule, we have that f'(x) = exp(ln(x)) * d/dx(ln(x)) = x * d/dx(ln(x)). But since f(x) = x, we must also have f'(x) = 1. Thus x * d/dx(ln(x)) = 1, or d/dx(ln(x)) = 1/x.
 
Muzza said:
Maybe something like this... Let f(x) = exp(ln(x)), where exp(x) = e^x. Using the chain rule, we have that f'(x) = exp(ln(x)) * d/dx(ln(x)) = x * d/dx(ln(x)). But since f(x) = x, we must also have f'(x) = 1. Thus x * d/dx(ln(x)) = 1, or d/dx(ln(x)) = 1/x.

Thank you Muzza! That was beautiful! How about my second problem?
 

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