Problem 59 - Algebra 2

Question

Solve each equation. Round to the nearest ten-thousandth. \(\ln \left(x^{2}+12\right)=\ln x+\ln 8\)

Step-by-Step Solution

Answer

The solutions are \(x = 6\) and \(x = 2\).

Step 1: Use Logarithmic Properties

The problem involves a logarithmic equation. Start by applying the property of logarithms that states \(\ln a + \ln b = \ln(ab)\). This allows us to rewrite the right side of the equation: \(\ln(x \cdot 8) = \ln(8x)\). The equation then becomes: \(\ln(x^2 + 12) = \ln(8x)\).

Step 2: Remove Logarithms

Since both sides of the equation are logarithms, we can equate the arguments (because the natural logarithm is a one-to-one function). This gives the equation \(x^2 + 12 = 8x\).

Step 3: Rearrange into a Quadratic Equation

Rearrange \(x^2 + 12 = 8x\) to form a standard quadratic equation. Move all terms to one side of the equation: \(x^2 - 8x + 12 = 0\).

Step 4: Solve the Quadratic Equation

Next, solve the quadratic equation: \(x^2 - 8x + 12 = 0\). Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -8\), and \(c = 12\).

Step 5: Apply the Quadratic Formula

Calculate the discriminant \(b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 12 = 64 - 48 = 16\). The roots are \(x = \frac{8 \pm \sqrt{16}}{2}\). Simplify to get \(x = \frac{8 \pm 4}{2}\). Thus, \(x = 6\) or \(x = 2\).

Step 6: Check for Valid Solutions

Both solutions need to be checked to ensure they are valid by substituting them back into the original equation.\(\ln((6)^2 + 12) = \ln 6 + \ln 8\)\(\ln(36 + 12) = \ln 6 + \ln 8\)\(\ln 48 = \ln 48\), which is valid. Similarly, for \(x = 2\),\(\ln((2)^2 + 12) = \ln 2 + \ln 8\)\(\ln(4 + 12) = \ln 2 + \ln 8\)\(\ln 16 = \ln 16\), which is also valid.

Key Concepts

Logarithmic Equations

Logarithmic equations involve variables inside logarithms. Solving them requires using the properties of logarithms to simplify and rearrange them.
In the given problem, we use the property \(\ln a + \ln b = \ln(ab)\). This helps combine the terms on the right side to \(\ln(8x)\).
When both sides have a natural logarithm, we can set the arguments equal to each other. That means \(x^2 + 12 = 8x\).

This simplification of logarithmic equations reduces complexity and enables easier solving. Logarithms help measure the power to which a base must be raised to produce a number. Natural logarithms are a specific type of logarithm with base \(e\).
Understanding these properties is crucial for handling logarithmic equations.

Quadratic Equation

A quadratic equation is one of the simplest forms of polynomial equations and comes in the standard form of \(ax^2 + bx + c = 0\).
Our solution demonstrates this by rearranging \(x^2 + 12 = 8x\) into the quadratic form \(x^2 - 8x + 12 = 0\).

In quadratic equations:

• \(a\) is the coefficient of \(x^2\),
• \(b\) is the coefficient of \(x\), and
• \(c\) is the constant term.

Quadratic equations are important as they can describe various real-world scenarios, like projectile motion or area of land. Understanding their structure is key to predicting behavior described by such equations.

Quadratic Formula

The quadratic formula is a tool for finding solutions, or roots, of quadratic equations. Given the equation \(ax^2 + bx + c = 0\), the formula is:\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
In our solution, we identify \(a = 1\), \(b = -8\), and \(c = 12\).
The discriminant, \(b^2 - 4ac\), is essential; it determines the number and type of roots.

If the discriminant is:

• positive, we have two real and distinct roots,
• zero, one real and repeated root,
• negative, two complex roots.

For our problem, the discriminant is 16, which is positive. Thus, we have two real solutions: 6 and 2. The quadratic formula is versatile, applicable in complex scenarios where factoring might be impractical.

Natural Logarithm

A natural logarithm, denoted by \(\ln\), measures the power needed to raise \(e\), a natural constant approximately equal to 2.718, to obtain a number.
It is a specific type of logarithm widely used in calculus, exponential growth models, and scientific calculations.

Natural logarithms have unique characteristics that make them suitable in many equations:

• The derivative of \(\ln x\) is \(\frac{1}{x}\), fitting calculus applications.
• They occur naturally in continuous growth processes.

The equation \(\ln(x^2 + 12) = \ln(8x)\) directly uses natural logarithms to reduce complexity and logically derive solutions. This demonstrates the invaluable role of \(\ln\) in simplifying mathematical expressions by leveraging its unique properties and inherent simplicity.