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Chapter 4: Problem 115

Find the approximate solution to each equation. Round to four decimal places. $$e^{2 x}=3$$

Short Answer

Expert verified
x \approx 0.5493

Step by step solution

01

Isolate the exponential term

The given equation is \(e^{2x} = 3\). To solve for \(x\), first isolate the exponential term on one side of the equation, which is already done in this case.
02

Apply natural logarithm

Apply the natural logarithm (\(\text{ln}\)) to both sides of the equation to get rid of the exponent: \(\ln(e^{2x}) = \ln(3)\).
03

Use logarithm properties

Utilize the property of logarithms that \(\ln(e^a) = a\) to simplify the left side of the equation: \(2x = \ln(3)\).
04

Solve for x

Divide both sides of the equation by 2 to isolate \(x\): \(x = \frac{\ln(3)}{2}\).
05

Calculate the value

Use a calculator to approximate the value of \(x\). First, find the natural logarithm of 3, which is approximately 1.0986. Then, divide by 2 to get \(x \approx \frac{1.0986}{2} \approx 0.5493\).

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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 \(\text{ln}\), is a special way of expressing logarithms. It uses the base of the natural exponential function, which is the number \(e\).\(e\) is approximately equal to 2.71828. Natural logarithms help simplify expressions involving exponential functions. For example, in the original problem, taking the natural logarithm of both sides of the equation lets us get rid of the exponent. This simplifies solving for our variable, \(x\). By remembering that \(\text{ln}(e^a) = a\), we can handle many exponential equations effectively.
Logarithm Properties
Logarithms have several useful properties that simplify solving equations. One main property is the power rule: \(\text{ln}(a^b) = b \cdot \text{ln}(a)\). Also, the natural logarithm of \(e^a\) is simply \(a\). \(\text{ln}(e^a) = a\). This is useful in solving exponential equations. In the given solution, using the property \( \text{ln}(e^{2x}) = 2x\) helped to transform a complex exponential equation into a simple linear form. From there, it was straightforward to solve for \(x\) by basic algebra.
Approximations
Approximations are often used in mathematics to give practical solutions when exact values are not necessary or are difficult to determine. In the exercise, once we found that \(x = \frac{\text{ln}(3)}{2}\), we used a calculator to find the approximate value of \(\text{ln}(3) \approx 1.0986\). Dividing this by 2 gives \( x \approx 0.5493\). Such approximations are essential in real-world applications and provide a useful answer when exact solutions are impractical or cumbersome.

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Most popular questions from this chapter

Solve \(\log _{x}(9.8)=2.4 .\) Round to four decimal places.

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