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Chapter 10: Problem 194

In the following exercises, solve each logarithmic equation. \(\ln x=5\)

Short Answer

Expert verified
The solution is \( x \approx 148.413 \).

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\ln x=5\). Recall that \(\ln x\) represents the natural logarithm of \(\x\), which is the same as the logarithm base \( \text{e} \). This step explains that we want to isolate \( x \).
02

Rewrite the logarithmic equation in exponential form

To solve for \( x \), rewrite the equation in exponential form. The natural logarithm \( \ln x = 5 \) can be written as \( e^5 = x \).
03

Solve for x

Now that we have \( e^5 = x \), we simply need to determine the value of \( e^5 \). Using a calculator or known values, \( e^5 \) is approximately \( 148.413 \). Thus, \( x \) is approximately \( 148.413 \).

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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, denoted as \(\text{ln} \), is a logarithm to the base \(\text{e}\). The number \(\text{e}\) is an irrational constant approximately equal to 2.71828. Natural logarithms are widely used in mathematics, particularly in calculus and complex analysis. They help in solving equations involving exponential growth or decay.

Here's a simple way to understand the natural logarithm:
  • It answers the question: 'To what power must e be raised, to get a given number?' For example, \(\text{ln}(x) = y\) means that \(\text{e}\) must be raised to the power of \(\text{y}\) to get \(\text{x}\).
  • In the exercise, \(\text{ln}(x) = 5\) means placing \(\text{x}\) in the form of \(\text{e}\) raised to some power results in 5.
This concept is vital for simplifying and solving exponential equations and is a fundamental part of higher mathematics.
exponential form
Rewriting logarithmic equations in exponential form makes solving them more manageable. Exponential form is where the expression is rewritten with a base and an exponent. For natural logarithms, the base is always \(\text{e}\).

In the given problem, \(\text{ln}(x) = 5\) can be rewritten as an exponential form:
  • Here, \(\text{ln}(x) = 5\) implies that \(\text{e}^5 = x\).
  • This means that 5 is the power to which \(\text{e}\) must be raised to result in \(\text{x}\).
Once in this form, it is straightforward to find the value of \(\text{x}\), simply by calculating \(\text{e}\) raised to the power of 5. Use a calculator to approximate, resulting in \(\text{x} \approx 148.413\). Converting logarithmic expressions to exponential form is often the key step in solving these equations.
math problem solving
Solving logarithmic equations involves a series of methodical steps. Let's break down the process once more:
  • Step 1: Understand the logarithmic equation: Recognize the form and base of the logarithm. For example, \(\text{ln}(x) = 5 \) signifies that \(\text{ln}\) is a natural logarithm with base \(\text{e}\).
  • Step 2: Rewrite the equation in exponential form: Converting \(\text{ln}(x) = 5 \) to \(\text{e}^5 = x \) aligns your equation with properties of exponents.
  • Step 3: Solve for x: Calculate the exponential value. Here, \(\text{e}^5 \) is approximately \(\text{148.413}\).
By approaching logarithmic equations with these straightforward steps, you can solve them efficiently.
Learning to approach problems systematically enhances your mathematical thinking and problem-solving skills. Always break down the problem, apply your knowledge of properties and operations, and you'll often find solutions become clear.

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Most popular questions from this chapter

In the following exercises, solve each equation. \(\log _{6}(x+1)-\log _{6}(4 x+10)=\log _{6} \frac{1}{x}\)

In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible. \(\log x^{-2}\)

In the following exercises, solve each equation. \(3^{3 x+1}=81\)

In the following exercises, solve for \(x\). \(\log (x-1)-\log (x+3)=\log \frac{1}{x}\)

In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm. \(\log _{3} 42\)
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