Is e^-ln(x) equal to 1/x? | Homework Statement

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SUMMARY

e^-ln(x) is definitively equal to 1/x. This conclusion is derived from the property of logarithms where -ln(x) can be rewritten as ln(1/x). Consequently, applying the exponential function results in e^-ln(x) = e^ln(1/x), which simplifies to 1/x. This mathematical identity is crucial for understanding the relationship between exponential and logarithmic functions.

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  • Understanding of logarithmic properties, specifically the relationship between natural logarithms and exponentials.
  • Familiarity with the exponential function e^x and its properties.
  • Basic algebra skills for manipulating equations and expressions.
  • Knowledge of real numbers and their properties in mathematical contexts.
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  • Study the properties of logarithms, focusing on the change of base and negative logarithms.
  • Learn about the exponential function and its applications in calculus.
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Homework Statement



is e^-ln(x) the same as 1/x ??

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The Attempt at a Solution

 
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What do you think?
 


What do you know about numbers and exponents to those numbers?

Let t be any real number. What can you do with e^{-t} ?
 


Well perhaps a way of verifying your answer would be to note -ln(x) = ln(1/x); therefore, e^-ln(x) = e^ln(1/x). Another way is to note e^-ln(x) = 1/e^ln(x).
 


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