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Chapter 10: Problem 44

Solve each equation. Round to the nearest ten-thousandth. $$ e^{3 x}=4 $$

Short Answer

Expert verified
The solution is approximately \( x = 0.4621 \).

Step by step solution

01

Understand the problem

We are given the equation \( e^{3x} = 4 \) and are tasked with solving for \( x \). The expression involves the natural exponential function \( e \). Our goal is to isolate \( x \) in the equation.
02

Apply the natural logarithm

To isolate \( x \), we take the natural logarithm (ln) of both sides of the equation. This gives us:\[ \ln(e^{3x}) = \ln(4) \]This step uses the property that \( \ln(e^y) = y \).
03

Simplify using logarithmic identities

Using the property \( \ln(e^{3x}) = 3x \), we simplify the left side of the equation to obtain:\[ 3x = \ln(4) \]
04

Solve for x

To solve for \( x \), divide both sides of the equation by 3:\[ x = \frac{\ln(4)}{3} \]Calculate \( \ln(4) \) using a calculator, and then divide by 3 to find \( x \).
05

Calculate and round the result

First, find \( \ln(4) \approx 1.386294361 \). Then, divide by 3 to find \( x \approx 0.46209812 \).Round this result to the nearest ten-thousandth to get \( x \approx 0.4621 \).

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Key Concepts

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

Natural Logarithm
The natural logarithm, often denoted as \( \ln(x) \), is a logarithm with the base \( e \), where \( e \) is an irrational constant approximately equal to 2.71828. The natural logarithm is the inverse of the exponential function.
  • Inverse Relationship: This means if \( e^y = x \), then \( \ln(x) = y \).
  • Key Property: An important property used in calculations is \( \ln(e^y) = y \). This identity plays a crucial role in simplifying exponential equations, as seen in the original exercise.
By applying the natural logarithm to both sides of an equation like \( e^{3x} = 4 \), we employ its inverse nature to simplify and solve for variables. This simplification is a fundamental step in tackling exponential equations.
Rounding Numbers
Rounding numbers is the process of adjusting the digits of a number to make it simpler, keeping it as close as possible to the original. In mathematics, rounding is used to attain a specified level of precision, which is particularly useful for approximating irrational numbers or when an exact value is unnecessary.
  • Precision: The problem requires rounding the result to the nearest ten-thousandth, which has four decimal places.
  • Rules: Generally, if the digit following the desired precision level is 5 or higher, we round up. If not, we round down.
For example, when we calculate \( x \approx 0.46209812 \), the digit in the fifth place is 9, so we round \( 0.46209812 \) to \( 0.4621 \), achieving the necessary precision.
Logarithmic Identities
Logarithmic identities are mathematical rules that make it easier to work with logarithms. They simplify expressions and provide a smooth transition from exponential to logarithmic forms. Here are a few important logarithmic identities:
  • Power Rule: \( \ln(a^b) = b \cdot \ln(a) \). This rule is instrumental when dealing with powers inside a logarithm.
  • Product Rule: \( \ln(a \cdot b) = \ln(a) + \ln(b) \).
  • Quotient Rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
In our example, the Power Rule helps us rewrite and simplify the expression \( \ln(e^{3x}) \) to \( 3x \), because \( \ln(e^y) = y \). These identities assist in breaking down complex expressions into manageable parts, making calculations more straightforward.

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Most popular questions from this chapter

PREREQUISITE SKILL Solve each system of equations. $$ \begin{array}{l}{4 x+y=14} \\ {4 x-y=10}\end{array} $$

Graph each system of equations. Use the graph to solve the system. $$ \begin{array}{ll}{\text { a. } 4 x-3 y=0} & {\text { b. } y=5-x^{2}} \\\ {x^{2}+y^{2}=25} & {y=2 x^{2}+2}\end{array} $$

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ -x y-2 x-3 y+6=0 $$

Each equation is of the form \(A x^{2}+B x y+C y^{2}+D x+\) \(E y+F=0 .\) Identify the values of \(A, B,\) and \(C\). $$ 2 x^{2}+3 x y-5 y^{2}=0 $$

The orbit of Pluto can be modeled by the equation \(\frac{x^{2}}{39.5^{2}}+\frac{y^{2}}{38.3^{2}}=1,\) where the units are astronomical units. Suppose a comet is following a path modeled by the equation \(x=y^{2}+20\) Will the comet necessarily hit Pluto? Explain.
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