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Chapter 4: Problem 4

If \(e^{-x}=3.2,\) write \(x\) in terms of the natural logarithm.

Short Answer

Expert verified
x = -\( \ln(3.2) \)

Step by step solution

01

- Understand the Problem

The given equation is an exponential equation: \[ e^{-x} = 3.2 \]. The goal is to express the variable \(x\) in terms of the natural logarithm.
02

- Take the Natural Logarithm on Both Sides

To solve for \(x\), take the natural logarithm (\( \ln \)) of both sides of the equation: \[ \ln(e^{-x}) = \ln(3.2) \].
03

- Simplify the Left Side Using Logarithm Properties

Apply the logarithm property \( \ln(a^b) = b \ln(a) \) to the left side of the equation: \[ -x \ln(e) = \ln(3.2) \]. Since \( \ln(e) = 1 \), this simplifies to: \[ -x = \ln(3.2) \].
04

- Solve for x

Isolate \(x\) by multiplying both sides by -1: \[ x = -\ln(3.2) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

exponential equations
An exponential equation is an equation in which the variable appears in an exponent. They are often in the form:
  • \(a^x = b\)
    • Exponential equations are a common way to describe phenomena in real life, like population growth, radioactive decay, or interest calculations.
    In the exercise, the equation was \(e^{-x} = 3.2\), where \(e\) is the base of the natural logarithm (approximately 2.718).
    To solve such equations, we often use logarithms, especially when the base is the natural logarithm base \(e\).
    logarithm properties
    Logarithm properties are rules that help simplify the operations involving logarithms. These properties are essential in solving exponential equations.
    Here are some key logarithm properties used in the exercise:
    • Power Rule: \(\ln(a^b)=b \ln(a)\): This property allows us to move an exponent in a logarithm to the front. We used this to write \(\ln(e^{-x})\) as \(-x \ln(e)\).
    • \(\ln(e) = 1\): Since the natural logarithm of the base of natural logarithms \(e\) is 1, we use this to simplify \(-x \ln(e)\) to \(-x\).
      • By leveraging these properties, we can transform complex equations into more manageable forms.
      solving for variables
      To solve for a variable in an equation means to isolate the variable on one side of the equation. This process involves using inverse operations and properties of equality.
      In the given exercise, the goal was to solve for \(x\). We started by taking the natural logarithm of both sides of the equation: \(\ln(e^{-x}) = \ln(3.2)\).
      • Using the power rule of logarithms, we simplified the left side: \(-x \ln(e)\).
      • Since \(\ln(e) = 1\), it becomes \(-x = \ln(3.2)\).
      • Finally, isolating \(x\) requires multiplying both sides by -1: \(x = -\ln(3.2)\).
      These steps highlight the systematic approach to isolating and solving for the variable in an exponential equation.

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      Most popular questions from this chapter

      Solve the given equation for \(x .\) $$3 \ln x-\ln 3 x=0$$

      Differentiate the following functions. $$y=\frac{x+1}{e^{x}}$$

      Graph the function \(f(x)=2^{x}\) in the window \([-1,2]\) by \([-1,4],\) and estimate the slope of the graph at \(x=0\).

      (a) Use the fact that \(e^{4 x}=\left(e^{x}\right)^{4}\) to find \(\frac{d}{d x}\left(e^{4 x}\right) .\) Simplify the derivative as much as possible. (b) Take an approach similar to the one in (a) and show that, if \(k\) is a constant, \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x}.\)

      Solve the given equation for \(x .\) $$\ln (x+1)-\ln (x-2)=1$$
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