Problem 66 - Precalculus

Question

Solve the logarithmic equations. Round your answers to three decimal places. $$\ln (4 x)+\ln (2+x)=2$$

Step-by-Step Solution

Answer

The positive solution to the equation is approximately \(x = 0.405\).

Step 1: Apply Logarithm Addition Property

The equation given is \(\ln (4x) + \ln (2+x) = 2\). According to the properties of logarithms, \(\ln a + \ln b = \ln (ab)\). Use this property to combine the logarithms: \(\ln ((4x)(2+x)) = 2\).

Step 2: Exponentiate to Remove Logarithm

To eliminate the natural log, exponentiate both sides of the equation: \(e^{\ln ((4x)(2+x))} = e^2\). This simplifies to: \((4x)(2+x) = e^2\).

Step 3: Simplify and Expand

Expand the expression: \(4x(2+x) = 8x + 4x^2\). Therefore, the equation becomes \(4x^2 + 8x = e^2\).

Step 4: Rearrange Into Standard Quadratic Form

Rearrange the equation into the standard quadratic form \(ax^2 + bx + c = 0\): \(4x^2 + 8x - e^2 = 0\).

Step 5: Solve the Quadratic Equation

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\). Here, \(a = 4\), \(b = 8\), and \(c = -e^2\). Substitute these values into the formula: \[x = \frac{-8 \pm \sqrt{8^2 - 4(4)(-e^2)}}{2(4)}\]Simplify under the square root and solve for \(x\).

Step 6: Calculate and Evaluate Discriminant

Calculate the discriminant: \(b^2 - 4ac = 64 + 16e^2\). This value is positive, meaning there are two real solutions.

Step 7: Solve for Real Roots

Calculate the roots using the quadratic formula:\[x = \frac{-8 \pm \sqrt{64 + 16e^2}}{8}\]Compute both possible solutions for \(x\) and round to three decimal places.

Step 8: Evaluate and Select Solutions

Since \(x\) must be a positive value (as it is inside a logarithm), discard negative solutions. Compute the positive solution from the quadratic, round it, and this will be the final solution.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations, which typically take the form \( ax^2 + bx + c = 0 \). This formula allows you to find the roots of the equation, or essentially the values of \( x \) that make the equation true. The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works step-by-step:

• The expression \( \pm \sqrt{b^2 - 4ac} \) is known as the discriminant. The value of the discriminant helps determine the type and number of solutions.
• If it is positive, as in our example, the quadratic has two distinct real roots.
• If it equals zero, there is exactly one real root (the roots are identical).
• If it is negative, there are no real roots, only complex ones.

Calculating these accurately involves carefully following the arithmetic steps, squaring \( b \), multiplying and subtracting \( 4ac \), and ensuring you've handled any negative values correctly. Then, plugging these into the formula gives you the potential solutions. Finally, always check the context of your problem to decide which solutions apply; for logarithmic equations, only positive solutions are valid.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base of \( e \), where \( e \) is approximately equal to 2.71828. It's a fundamental mathematical function often seen in calculus and exponential growth problems. In the context of the equation \( \ln (4x) + \ln (2+x) = 2 \), we encounter properties of the natural logarithm.A key property is how it translates multiplication inside the argument into addition outside, which is expressed as:\[ \ln(a \cdot b) = \ln a + \ln b \]Here's how it is applied:

• In combining \( \ln (4x) + \ln (2+x) \), they simplify to \( \ln((4x)(2+x)) \).
• The natural logarithm is employed to simplify the equation and then exponentiating both sides eliminates the logarithm, presenting us with a quadratic to solve.
In natural logarithms, the concept revolves around the continuous compounding rate, often used in growth and decay models. Transforming the logarithm equation into a simpler format through properties of logarithms can significantly ease solving complex problems.

Properties of Logarithms

The properties of logarithms are mathematical rules that allow you to manipulate logarithmic expressions more conveniently. They're incredibly useful when solving logarithmic equations like \( \ln(4x) + \ln(2+x) = 2 \). Here are some essential properties:

• Product Rule: \( \ln a + \ln b = \ln(ab) \) - Used to combine logs with addition.
• Quotient Rule: \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \) - Useful for subtracting logs.
• Power Rule: \( \ln(a^b) = b \cdot \ln a \) - Helps bring down exponents.

These properties are vital in simplifying expressions and solving equations. For example, the product rule combines separate logs into a single logarithm, which can be exponentiated and solved, simplifying our logarithmic equation to a quadratic equation. Leveraging these properties effectively can turn complex problems into simpler, more manageable forms, paving the way for a solution. In practical terms, understanding and practicing these properties are crucial for efficiently working through logarithmic problems.