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Chapter 3: Problem 100

Solve each equation. $$\ln 3-\ln (x+5)-\ln x=0$$

Short Answer

Expert verified
After simplifying the equation, interpreting the logarithmic terms, and solving for the roots, make sure that the solutions obtained are positive and valid for the original equation.

Step by step solution

01

Simplify the Equation

By applying the properties of logarithms, specifically that \(\ln a - \ln b = \ln \left( \frac{a}{b} \right)\), the equation can be rewritten as \(\ln \left( \frac{3}{(x+5)x}\right) = 0\).
02

Interpret the Logarithmic Equation

When the natural logarithm of a number equals 0, that number must be 1 because exponentiating \(e^0 = 1\). This implies \(\frac{3}{(x+5)x} = 1\).
03

Solve for \(x\)

To solve the equation \(\frac{3}{(x+5)x} = 1\), it can be rearranged to get the quadratic equation \(x^2 + 5x - 3 = 0\). This can be solved by applying the quadratic formula \(x= \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).
04

Evaluate the Roots

By substituting \(a = 1\), \(b = 5\) and \(c = -3\) into the quadratic formula, two roots can be obtained. One should validate these roots to ensure they're valid for the original logarithmic equation, since the domain of the natural logarithm is \(x > 0\).

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Key Concepts

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

Properties of Logarithms
Understanding the properties of logarithms is key to solving logarithmic equations effectively. One of the essential properties is the difference of logarithms, expressed as \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \).
This rule allows us to simplify expressions by combining multiple logarithmic terms into one鈥攁n important step in the original exercise.
  • When given an equation like \( \ln 3 - \ln (x+5) - \ln x = 0 \), we use this property to rewrite the expression into a single logarithm: \( \ln \left( \frac{3}{(x+5)x} \right) = 0 \).
  • The reason we pursue this simplification is that a single logarithm makes it easier to apply other mathematical operations and identify solutions.
Grasping this and other logarithmic properties will greatly help in tackling not only textbook problems like the one above but also real-world calculations involving exponential growth or decay.
Quadratic Equation
Quadratic equations play a crucial role when solving certain logarithmic equations, much like the one we have here. By simple rearrangement or transformation, we eventually encounter a quadratic equation: \( x^2 + 5x - 3 = 0 \).
Solving quadratic equations typically involves either factoring, completing the square, or using the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
  • In this context, the coefficients \( a = 1 \), \( b = 5 \), and \( c = -3 \) are plugged into the formula to find potential solutions for \( x \).
  • This equation reflects the algebraic manipulation required once the logarithmic equation has been simplified.
Using the quadratic formula is a structured way to get solutions even if the equation cannot be factored simply.
Domain of Logarithmic Functions
The domain of logarithmic functions is a critical aspect to be aware of when solving logarithmic equations. It establishes constraints on acceptable solutions.
Logarithms, in particular with the natural log, require their arguments to be positive: \
  • For \( \ln (x+5) \) and \( \ln x \) in the problem, both \( x+5 \) and \( x \) must be greater than zero.
  • This translates to \( x > 0 \), ensuring the values hold true within the context of the original equation.
After solving the quadratic equation, it's necessary to verify that obtained solutions adhere to this domain.
This ensures no extraneous solutions are included, leading to an accurate and valid resolution of the problem.

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