Logarithmic function defined as an integral

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SUMMARY

The discussion focuses on proving the properties of the logarithmic function defined as an integral, specifically Log(x) = ∫[1,x] (1/t) dt. The user is successfully applying the product rule but encounters difficulties with the quotient rule, Log(a/b) = Log(a) - Log(b). A suggested approach involves using the Chain Rule to derive the properties of logarithms, leading to the conclusion that log(a/x) can be expressed as -log(x) + c, where c is a constant determined by setting log(1) to zero.

PREREQUISITES
  • Understanding of integral calculus, specifically definite integrals.
  • Familiarity with logarithmic functions and their properties.
  • Knowledge of the Chain Rule in differentiation.
  • Ability to manipulate algebraic expressions involving logarithms.
NEXT STEPS
  • Study the proof of Log(a) + Log(b) = Log(ab) using integral definitions.
  • Explore the application of the Chain Rule in calculus for logarithmic functions.
  • Investigate the derivation of logarithmic properties from first principles.
  • Learn about the implications of setting log(1) = 0 in logarithmic functions.
USEFUL FOR

Students and educators in calculus, mathematicians interested in logarithmic properties, and anyone looking to deepen their understanding of integral definitions of logarithmic functions.

JMR_2413
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I'm attempting to prove the rules for the logarithmic function using the Integral definition: Log(x)=[1,x]∫1/t dt. I think I am alright with the product rule but I'm struggling with the quotient rule: i.e. Log(a/b)=Log(a)-Log(b). I believe that I'm having trouble breaking up the Integral correctly. Any help would be appreciated!
 
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If you have already proved Log(a) + Log(b) = Log(ab) from the integral definition
and you want to prove Log(a/b) + Log(b) = Log(a),
just replace a by a/b everywhere in your proof.
 
JMR_2413 said:
I'm attempting to prove the rules for the logarithmic function using the Integral definition: Log(x)=[1,x]∫1/t dt. I think I am alright with the product rule but I'm struggling with the quotient rule: i.e. Log(a/b)=Log(a)-Log(b). I believe that I'm having trouble breaking up the Integral correctly. Any help would be appreciated!

Try using the Chain Rule.

d/dxlog(a/x) = (x/a)(-a/x^2) = -1/x so log(a/x) = -log(x) + c.

Now calculate c.

The Chain Rule can be used to find all of the properties of the log once one sets log(1) to equal zero.
 

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