Problem 81 - Elementary and Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ 2 \log _{2} x=3+\log _{2}(x-2) $$

Step-by-Step Solution

Answer

The exact solution is \(x = 4\).

Step 1: Rewrite the equation with a common logarithmic base

The given equation is \(2 \log_{2} x = 3 + \log_{2}(x-2)\). Start by expressing everything in terms of logarithms first:\[ 2 \log_{2} x - \log_{2}(x-2) = 3 \]

Step 2: Apply logarithmic properties

Use the property of logarithms that states \(a \log_{b} c = \log_{b} c^{a}\) and \(\log_{b} a - \log_{b} c = \log_{b}\left(\frac{a}{c}\right)\):\[ \log_{2} \left(\frac{x^2}{x-2}\right) = 3 \]

Step 3: Convert logarithmic equation to an exponential equation

Convert the logarithmic equation to an exponential form using the definition \(\log_{b} a = c \Rightarrow b^{c} = a\):\[ \frac{x^2}{x-2} = 2^3 \]

Step 4: Simplify and solve the equation

Since \(2^3 = 8\), multiply both sides by \((x-2)\) to eliminate the fraction:\[ x^2 = 8(x-2) \]Expand and rearrange the equation:\[ x^2 = 8x - 16 \]\[ x^2 - 8x + 16 = 0 \]

Step 5: Solve the quadratic equation

Solve the quadratic equation \(x^2 - 8x + 16 = 0\). This equation can be solved using factorization or the quadratic formula.Since \((x-4)^2 = x^2 - 8x + 16\), it is a perfect square trinomial:\[ (x-4)^2 = 0 \]This implies that \(x = 4\).

Step 6: Verify the solution

Check that the solution \(x = 4\) satisfies the original equation. Substitute back into the original equation:\[ 2 \log_{2} 4 = 3 + \log_{2}(4 - 2) \]Since \(\log_{2} 4 = 2\), it simplifies to:\[ 2 \times 2 = 3 + 1 \]\[ 4 = 4 \]The solution verifies correctly.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and have the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The degree of the polynomial is two because of the term \(x^2\).
Understanding how to solve quadratic equations is crucial because they frequently appear in various mathematical contexts, including physics and engineering.

• **Factoring**: Involves expressing the quadratic in terms of its roots. This is useful when the equation can be written as a product of two linear expressions, such as \((x-p)(x-q) = 0\).
• **Quadratic Formula**: Given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It works for all quadratics and provides an exact solution when factoring is not straightforward.
• **Completing the Square**: A method that involves converting the quadratic into a perfect square trinomial, making it easier to solve.

For instance, in the example \((x-4)^2 = x^2 - 8x + 16\), we recognized this as a perfect square trinomial, resulting in the solution \(x = 4\). This simplification demonstrates the elegance and efficiency of recognizing special patterns within quadratics.

Logarithm Properties

Logarithms are inverses of exponential functions, effectively turning multiplication into addition. They are denoted as \(\log_b x\), where \(b\) is the base and \(x\) is the argument. Understanding the properties of logarithms simplifies many complex-looking problems and is essential for solving logarithmic equations.

• **Product Rule**: \(\log_b (xy) = \log_b x + \log_b y\). This property allows the multiplication of two arguments within a logarithm to become an addition operation.
• **Quotient Rule**: \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\). This is crucial for simplifying expressions where division is involved.
• **Power Rule**: \(\log_b (x^a) = a \cdot \log_b x\). It simplifies the expression if the argument of the logarithm is raised to a power.

In the original problem, we applied these properties to transform the logarithmic equation \(2 \log_2 x = 3 + \log_2(x-2)\) into a simpler form \(\log_2 \left(\frac{x^2}{x-2}\right) = 3\), showing how logarithmic properties help manage complex algebraic transformations.

Exponential Equations

Exponential equations involve variables in exponents and are characterized by the form \(a^x = b\), where \(a\) and \(b\) are constants. These equations are essential in modeling a wide range of real-world phenomena, such as population growth and radioactive decay. They are solved by applying the inverse operation of taking the logarithm or using properties of exponents.

• **Changing to Logarithms**: Exponential equations can often be solved by transforming them into logarithmic form. Using the relationship \(a^x = b \Rightarrow x = \log_a b\), we can find the exponential value.
• **Aligning Bases**: Another method involves expressing each side to the same base, which makes direct comparison and further simplification possible.
• **Understanding Base Exponentials**: Recognize that \(a^x\) represents repeated multiplication and that logarithms undo this process.

In our example, through the transformation of \(\log_2 \left(\frac{x^2}{x-2}\right) = 3\) into its exponential counterpart \(\frac{x^2}{x-2} = 2^3\), we effectively used exponential principles to simplify the equation to a solvable format, ultimately leading to the quadratic equation \(x^2 - 8x + 16 = 0\).