Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, solve each equation for \(x\). \(\ln (7)+\ln \left(2-4 x^{2}\right)=\ln (14)\)

Step-by-Step Solution

Answer

The solution is \( x = 0 \).

Step 1: Simplify the Equation Using Logarithm Properties

Combine the logarithms on the left side of the equation using the property \( \ln(a) + \ln(b) = \ln(ab) \). This results in \( \ln(7 \cdot (2 - 4x^2)) = \ln(14) \).

Step 2: Use the Property of Equality for Logarithms

Because the natural logarithms are equal, their arguments must be equal. Set the arguments equal: \( 7 \cdot (2 - 4x^2) = 14 \).

Step 3: Simplify the Equation

Distribute the 7: \( 14 - 28x^2 = 14 \).

Step 4: Isolate the x-Term

Subtract 14 from both sides of the equation: \( -28x^2 = 0 \).

Step 5: Solve for x

Divide both sides of the equation by -28: \( x^2 = 0 \).

Step 6: Find the Value of x

Take the square root of both sides to solve for \( x \): \( x = 0 \).

Key Concepts

Logarithmic Equations

Logarithmic equations often involve solving for a variable inside a logarithm. These types of equations are essential in algebra as they allow you to work with exponentially growing quantities. Typically, a logarithmic equation can be simplified using various properties of logarithms to isolate the variable. These equations often involve conditions where the arguments of the logarithms must be positive, as logarithms are undefined for non-positive numbers. Understanding and handling these conditions correctly is crucial for solving logarithmic equations accurately.

In the exercise problem, you had to deal with a natural logarithmic equation of the form: \[ \ln(7) + \ln(2 - 4x^2) = \ln(14) \] The goal was to simplify and find the value of \(x\). By using the properties of logarithms, such as combining logs with the same base, you simplify equations and create conditions where variables can be isolated and solved.

Properties of Logarithms

The properties of logarithms are powerful tools in simplifying and solving equations involving logs. These properties include:

• Product property: \( \ln(a) + \ln(b) = \ln(ab) \)
• Quotient property: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \)
• Power property: \( \ln(a^b) = b \cdot \ln(a) \)

These rules help in transforming complex expressions into simpler ones, which can be critical when equating equations or identifying domains.

In the exercise, you applied the product property to combine \(\ln(7) + \ln(2 - 4x^2)\) into one single logarithm: \[ \ln(7 \cdot (2 - 4x^2)) \] This manipulation reduced the equation to a more manageable form, showing how each property can untangle a logarithmic expression, paving the way to solution.

Solving Equations

Solving logarithmic equations often involves several steps that utilize mathematical properties and operations. Here's a quick guide you can follow:

• **Combine logarithms** using properties to simplify the equation.
• **Set the arguments equal** (when possible) to use the property that if \( \ln(a) = \ln(b) \), then \( a = b \).
• **Simplify the resulting equation** to isolate the variable. This might involve distributing terms, combining like terms, or moving parts of the equation to one side.
• **Solve for the variable** by using arithmetic operations or algebraic manipulations, such as factoring or applying the quadratic formula.

In the given exercise, after simplifying the logs, you equated their arguments and proceeded with normal algebraic manipulation. This included distributing constants, isolating terms with variables, and eventually solving for \(x\). Specifically, you used the approach: \[ 7 \cdot (2 - 4x^2) = 14 \] You simplified this to \( -28x^2 = 0 \), finally solving \(x\) by taking the square root of both sides, which yielded \(x = 0\). This systematic approach to breaking down the problem ensures precise solutions to logarithmic equations.