Problem 69 - Precalculus

Question

Solve the logarithmic equations. Round your answers to three decimal places. $$\ln \sqrt{x+4}-\ln \sqrt{x-2}=\ln \sqrt{x+1}$$

Step-by-Step Solution

Answer

The solution is approximately \( x = 3.646 \).

Step 1: Use Logarithm Properties

Recall the property of logarithms that states \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). Apply this property to the left side of the equation: \( \ln \left( \frac{\sqrt{x+4}}{\sqrt{x-2}} \right) = \ln \sqrt{x+1} \).

Step 2: Exponentiate Both Sides

Remove the logarithms by exponentiating both sides of the equation. This gives us: \( \frac{\sqrt{x+4}}{\sqrt{x-2}} = \sqrt{x+1} \).

Step 3: Simplify the Equation

Multiply both sides by \( \sqrt{x-2} \) to eliminate the fraction: \( \sqrt{x+4} = \sqrt{x+1} \cdot \sqrt{x-2} \).

Step 4: Square Both Sides

To eliminate the square roots, square both sides of the equation: \( x+4 = (x+1)(x-2) \).

Step 5: Expand and Simplify the Equation

Expand the right side: \( x+4 = x^2 - 2x + x - 2 \), which simplifies to \( x+4 = x^2 - x - 2 \).

Step 6: Rearrange into Quadratic Form

Bring all terms to one side to form a quadratic equation: \( x^2 - x - x - 2 - 4 = 0 \), leading to \( x^2 - 2x - 6 = 0 \).

Step 7: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -2 \), and \( c = -6 \). Calculate the discriminant: \( b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-6) = 4 + 24 = 28 \).

Step 8: Apply the Quadratic Formula

Apply the values in the formula: \( x = \frac{2 \pm \sqrt{28}}{2} \). Simplify to get \( x = 1 \pm \sqrt{7} \).

Step 9: Calculate Numerical Solutions

Evaluate \( x = 1 + \sqrt{7} \approx 3.646 \) and \( x = 1 - \sqrt{7} \approx -1.646 \).

Step 10: Check for Validity

Since a logarithm requires the argument to be positive, check potential solutions in the original expressions. \( x = 3.646 \) works as all terms in the original equation are positive, but \( x = -1.646 \) does not, as it makes square root terms negative.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial for solving equations efficiently. One key property to recall is the difference of logarithms: \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). This property allows us to combine or condense logarithmic expressions by subtracting or dividing their components. Such simplifications make equations easier to handle and solve.
Applying this to our exercise, the equation \( \ln \sqrt{x+4} - \ln \sqrt{x-2} = \ln \sqrt{x+1} \) is simplified using this property. We condense the left side, resulting in \( \ln \left( \frac{\sqrt{x+4}}{\sqrt{x-2}} \right) = \ln \sqrt{x+1} \). This reduction into a single logarithm on each side sets a clear path to eliminate the logarithm functions by exponentiation, streamlining the equation's complexity.
When working with logarithmic equations, always look for opportunities to employ these properties. They are powerful tools that transform logarithmic expressions into more manageable forms, paving the way for solving more complex equations.

Quadratic Equation

Once logarithmic properties bring the equation to a simpler form, it often translates into a quadratic equation upon further manipulation. Quadratic equations are polynomials of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Recognizing the transition into a quadratic form is key in solving the problem efficiently.
In our case, squaring both sides of the equation \( \sqrt{x+4} = \sqrt{x+1} \cdot \sqrt{x-2} \) leads to a quadratic equation. Expanding \((x+1)(x-2)\), followed by bringing all terms to one side, forms \( x^2 - 2x - 6 = 0 \).
Solving this quadratic equation involves finding roots that satisfy the equation. The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is a reliable method. By calculating the discriminant \( b^2 - 4ac \), you determine the nature of the roots—whether real or complex—and find the numerical solutions accordingly.

Solving Exponential Equations

While initially appearing as a logarithmic equation, the exercise transitions into a format involving exponentials and square roots, typical of exponential equations. Solution approaches here rely heavily on properties of exponents and logarithms.
After applying logarithmic properties and simplifications, the equation \( \frac{\sqrt{x+4}}{\sqrt{x-2}} = \sqrt{x+1} \) is exponentiated to remove the logarithms. When exponentiating, you effectively convert logarithmic equations into polynomial forms allowing for more straightforward algebraic manipulation.
Finally, after reaching solutions, always verify potential candidates by substituting them back into the original equation. This ensures they satisfy logarithmic restrictions, such as requiring their arguments to remain positive. Only the solutions meeting these conditions qualify as valid, highlighting how careful examination at each step secures the correct answer.