step 1 In this problem of exponential and logarithmic function we have to simplify the expression we ha

Question

$$ \ln \left(e^{\ln e}\right) $$

Name: Problem 5, Chapter 1: Calculus with Concepts in Calculus
Uploaded: 2022-04-06T11:52:18-08:00
Duration: 1 min 16 s
Channel: Prachita Kush
Description: ln(e^lne)

   $$
\ln \left(e^{\ln e}\right)
$$

Calculus with Concepts in Calculus

Denny Gulick, Robert… 6th Edition

Chapter 1, Problem 5

Instant Answer

Solved by Expert Prachita Kush

04/06/2022

Step 1

We know that $e^{\ln a}$ is equal to $a$ for any positive number $a$. Show more…

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$$ \ln \left(e^{\ln e}\right) $$

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Key Concepts

Natural Logarithm

The natural logarithm, denoted by ln, is a logarithm with base e. It is widely used in mathematics due to its unique calculus properties and the fact that it serves as the inverse function of the exponential function with the same base.

Exponential Function

The exponential function with base e, denoted as e^x, is a fundamental mathematical function characterized by its continuous and rapid growth. It is the inverse of the natural logarithm, meaning that applying the natural logarithm to an exponential function returns the exponent.

Inverse Relationship of Logarithms and Exponentials

The inverse relationship between the natural logarithm and the exponential function implies that ln(e^x) equals x. This property simplifies many expressions in algebra and calculus, allowing one to 'cancel out' the logarithm and the exponential when they appear together.

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