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Chapter 4: Problem 28

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

Short Answer

Expert verified
x = \sqrt{3}

Step by step solution

01

Simplify the Logarithms

Use the property of logarithms that states \(a \ln b = \ln b^a\). Simplify the equation \(\text{3} \ln x - \ln 3x = 0\) into \(\text{3} \ln x - \ln 3 - \ln x = 0\).
02

Combine the Logarithms

Combine the logarithms on the left side using the property \( \ln a - \ln b = \ln \left( \frac{a}{b} \right)\). This gives \(\text{3} \ln x - \ln 3x\) and can be written as \(\text{2} \ln x - \ln 3 = 0\).
03

Isolate \(x\)

Move \( \ln 3 \) to the right side of the equation: \(2 \ln x = \ln 3\).
04

Solve for \(x\)

Divide both sides by \(2\): \( \ln x = \ln 3 \left/ 2 \right.\). Exponentiate both sides to remove the logarithm: \(x = e^{\left( \frac{\ln 3}{2}\right)} = 3^{\left( \frac{1}{2}\right)} = \sqrt{3}\).

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Key Concepts

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

Properties of Logarithms
To solve logarithmic equations, it's crucial to understand the core properties of logarithms. Firstly, remember that \(a \ln b = \ln b^a\). This property allows you to move constants inside the logarithm. For instance, in the equation \(3 \ln x - \ln 3x = 0\), you can simplify \(3 \ln x\) to \ln x^3\. This gives a better approach to handle the equation. Another helpful property is \( \ln a - \ln b = \ln \left( \frac{a}{b} \right)\). Using these properties effectively transforms complex expressions into simpler ones.
Exponentiation
Exponentiation is another important concept in solving logarithmic equations. When you have an equation of the form \( \ln x = y\), you can exponentiate both sides to get rid of the logarithm, using \( e^y = x\). For example, in our solu tion, we convert \(2 \ln x = \ln 3\) to \( \ln x = \frac{ \ln 3}{2}\). Then, we exponentiate to find \( x = e^{\left( \frac{ \ln 3}{2}\right)}\), which simplifies to \(x = \sqrt{3}\). This step is necessary to isolate the variable and find the solution.
Solving Equations
The process of solving logarithmic equations involves simplifying, isolating, and solving for the variable. Let's break down these steps:
  • Simplify logarithmic expressions using properties of logarithms.
  • Combine logs wherever possible to condense the equation.
  • Isolate the logarithmic term by moving constants to the other side.
  • Exponentiate to remove the log and solve for the variable.
By following these systematic steps, you convert a complex logarithmic equation into a simpler one. For instance, in our example, after combining and isolating terms, we eventually exponentiate to find \( \ln x = \frac{ \ln 3}{2}\) and finally get \( x = \sqrt{3}\). Understanding this method will help tackle a variety of logarithmic problems with confidence.

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