Question

$\ln \left(2 e^{x+1}\right)-\ln \left(2 e^{x}\right)=1$ for all real numbers $x$

   $\ln \left(2 e^{x+1}\right)-\ln \left(2 e^{x}\right)=1$ for all real numbers $x$

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 6, Problem 8

Instant Answer

Solved by Expert Alex Roush

06/26/2021

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Step 1: We start with the given equation: \[\ln \left(2 e^{x+1}\right)-\ln \left(2 e^{x}\right)=1\] Show more…

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$\ln \left(2 e^{x+1}\right)-\ln \left(2 e^{x}\right)=1$ for all real numbers $x$

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Key Concepts

Properties of Logarithms

This concept involves the various rules governing logarithmic expressions, such as the quotient rule, which states that the difference of two logarithms with the same base is equal to the logarithm of the quotient of their arguments. Understanding these properties allows for the simplification and manipulation of logarithmic equations, making it easier to solve them.

Exponential Functions

Exponential functions, particularly those involving the natural base e, have properties that allow exponents to be split or combined, such as e^(a+b) = e^a · e^b. This trait is crucial when dealing with expressions that combine exponential and logarithmic functions, as it enables the cancellation or simplification of terms.

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