Question

Find $k$ such that $2^{-x / 5}=e^{k x}$ for all $x.$

   Find $k$ such that $2^{-x / 5}=e^{k x}$ for all $x.$

Calculus and Its Applications

Calculus and Its Applications

Larry J. Goldstein,… 14th Edition

Chapter 4, Problem 52

Instant Answer

Solved by Expert Linh Vu

10/27/2020

Step 1

This gives us: \[\ln(2^{-x / 5})=\ln(e^{k x})\] Show more…

Show all steps

lock

Thanks for your feedback!

Find $k$ such that $2^{-x / 5}=e^{k x}$ for all $x.$

Play audio

Feedback

Powered by NumerAI

Linh Vu and 50 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Exponential Functions

Exponential functions involve expressions where the variable appears in the exponent, such as a^(bx). In solving equations where exponential expressions are set equal, it is often necessary to compare and manipulate the exponents directly.

Change of Base and Logarithmic Conversion

Changing the base of exponential functions through logarithmic conversion is a key technique. It facilitates rewriting an expression like 2^(-x/5) in terms of the natural exponential function using the identity a^b = e^(b ln a), allowing for easier comparison of exponential expressions.

Equating Exponents

When two exponential expressions with the same base are equal for all values of the variable, their exponents must be equal. This concept is essential in solving equations where a rewritten exponential equation in base e leads to equating the exponent expressions, thereby determining unknown parameters.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later