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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
The solution for \( x \) is \( \mathbf{ \sqrt{3} } \).

Step by step solution

01

- Simplify the logarithmic expression

Start by applying the logarithmic property that allows combining logs: \[3 \ln(x) - \ln(3x) = 0\] Apply the property \( \ln(a/b) = \ln(a) - \ln(b) \) to the term \( \ln(3x) \): \[3 \ln(x) - ( \ln(3) + \ln(x) ) = 0\]
02

- Simplify the equation

Combine like terms: \[3 \ln(x) - \ln(3) - \ln(x) = 0 \]This simplifies to: \[2 \ln(x) - \ln(3) = 0\]
03

- Isolate the logarithmic term

Add \( \ln(3) \) to both sides: \[2 \ln(x) = \ln(3)\]
04

- Solve for \( x \)

Divide both sides by 2 to isolate \( \ln(x) \): \[\ln(x) = \frac{\backslash ln(3)}{2}\] Rewrite the equation in exponential form to solve for \( x \): \[x = \exp\backslash left( \frac{\ln(3)}{2} \right) \] This simplifies to: \[x = \sqrt{3}\]

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Key Concepts

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

logarithm properties
Logarithmic properties play a crucial role in simplifying and solving logarithmic equations. The key properties you'll often use include:
  • \textbf{Product Property}: \ \ ln(a) + ln(b) = ln(ab)
  • \textbf{Quotient Property}: \ \ ln(a) - ln(b) = ln\frac{a}{b}
  • \textbf{Power Property}: \ \ ln(a^b) = b ln(a)
These properties allow us to combine multiple logarithms into a single logarithmic expression or to split them apart for simplification. For instance, in the given exercise, we convert \( ln(3x) \) to \( ln(3) + ln(x) \), using the product property in reverse to separate the logarithmic terms. Understanding these properties is key to manipulating and solving logarithmic equations efficiently.
solving equations
Solving logarithmic equations involves a sequence of steps designed to isolate the variable. The general approach includes:
  • Applying Logarithmic Properties: Use properties like the product, quotient, and power rules to combine or simplify terms.
  • Isolating the Logarithmic Term: Move all logarithmic expressions to one side of the equation for easier manipulation.
  • Exponentiating Both Sides: Transform the logarithmic equation into an exponential one, which often makes it simpler to solve.
  • Simplifying: Solve for the variable after rewriting the expression in its exponential form.
For example, in the provided exercise:
  • First, we used the properties of logarithms to combine terms.
  • Then, we isolated the logarithmic term, moving everything else to the other side.
  • After that, we converted the log equation to its exponential form for final simplification, making it easy to solve for \( x \).
Consistently implementing these steps ensures a more straightforward path to the solution.
exponential functions
Exponential functions are the counterparts to logarithmic functions. They 'undo' logarithms and are integral when solving log equations. When given an equation like \( ln(x) = y \), we convert it to its exponential form as \( x = e^y \), where \( e \) is Euler's number (approximately 2.718). In the exercise, once we had \( ln(x) = \frac{ ln(3) }{ 2 } \), we converted it to exponential form, \( x = e^{ \frac{ ln(3) }{ 2 } } \), simplifying to \( x = \sqrt{3} \).
Recognizing when to apply the exponential function can help solve log equations quickly.
  • Identifying log-to-exponential transformations helps in converting complex log equations to more straightforward exponential ones.
  • Exponential Functions grow rapidly and are useful in diverse fields, from calculus to real-world applications like population growth and radioactive decay.
  • Being comfortable with exponential functions allows for smoother transitions in manipulating and solving logarithmic equations.

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