Proving the Inverse Relationship of ln(x) and e^x

  • Thread starter Thread starter appplejack
  • Start date Start date
Click For Summary
SUMMARY

The discussion centers on proving the equation ln(e^x) = x, which is established through the definition of the natural logarithm as the inverse function of the exponential function e^x. The proof can also be approached by recognizing that ln(e) equals 1, simplifying the equation to ln(e^x) = x * ln(e) = x. This confirms that the natural logarithm and the exponential function are inverses of each other.

PREREQUISITES
  • Understanding of natural logarithms and their properties
  • Familiarity with exponential functions, specifically e^x
  • Basic algebraic manipulation skills
  • Knowledge of the definition of inverse functions
NEXT STEPS
  • Study the properties of logarithmic functions, focusing on natural logarithms
  • Explore the concept of inverse functions in greater depth
  • Learn about the derivative of e^x and its relationship to ln(x)
  • Investigate applications of logarithmic and exponential functions in real-world scenarios
USEFUL FOR

Students studying calculus, mathematics educators, and anyone interested in understanding the relationship between logarithmic and exponential functions.

appplejack
Messages
43
Reaction score
0

Homework Statement


How do you prove ln(e^x) = x?


Homework Equations





The Attempt at a Solution


I just memorized this but want to know how I can prove this
 
Physics news on Phys.org
It's the definition of ln(x) that it is the inverse of e^x. If you want another way then ln(e^x)=x*ln(e). What's ln(e)?
 

Similar threads

Proving limits for roots and exponents
  • · Replies 13 ·
Replies
13
Views
2K
Solve the first order linear differential equation
  • · Replies 26 ·
Replies
26
Views
1K
How can we prove ##e^{ln x}= x## and ##e^-{ln(x+1)}= \frac 1 {x+1}##?
  • · Replies 5 ·
Replies
5
Views
1K
Find the equation of given curve in the form ##e^{3y}=f(x)##
  • · Replies 4 ·
Replies
4
Views
1K
Find the roots of the given hyperbolic equation
  • · Replies 4 ·
Replies
4
Views
1K
Finding the Inverse Function of tanh(x) in the Interval (-1,1)
  • · Replies 2 ·
Replies
2
Views
1K
Solve the given equation using the Lambert W function
  • · Replies 2 ·
Replies
2
Views
2K
Using Euler's formula to prove trig identities using "sum to product" technique
  • · Replies 4 ·
Replies
4
Views
2K
Proving function f(x) is a PDf given integral relationships
  • · Replies 1 ·
Replies
1
Views
2K
Improper integral convergence from 0 to 1
  • · Replies 13 ·
Replies
13
Views
3K