Problem 112 - A Graphical Approach to Precalculus with Limits

Question

Use any method (analytic or graphical) to solve each equation. $$\log x^{2}=(\log x)^{2}$$

Step-by-Step Solution

Answer

The solutions are \( x = 1 \) and \( x = 100 \).

Step 1: Rewrite the Equation

Start by rewriting the given equation \[ \log x^2 = (\log x)^2 \]Using the property of logarithms, such as \( \log a^b = b \cdot \log a \), rewrite the left-hand side as \( 2 \cdot \log x \):\[ 2 \cdot \log x = (\log x)^2 \]

Step 2: Introduce a New Variable

Substitute \( \log x = y \) to simplify the equation to \[ 2y = y^2 \] This substitution helps in solving the equation using algebraic methods.

Step 3: Rearrange the New Equation

Rearrange the equation to form a standard quadratic equation:\[ y^2 - 2y = 0 \]Factor the equation to find its solutions.

Step 4: Factor the Quadratic Equation

Factor the quadratic equation: \[ y(y - 2) = 0 \]This gives us two potential solutions: \( y = 0 \) or \( y = 2 \).

Step 5: Substitute Back to Original Variable

Substitute back \( \log x = y \) to solve for \( x \):- For \( y = 0 \): \( \log x = 0 \). It implies: \[ x = 10^0 = 1 \]- For \( y = 2 \): \( \log x = 2 \). It implies: \[ x = 10^2 = 100 \]

Step 6: Verify Solutions

Verify both solutions \( x = 1 \) and \( x = 100 \) in the original equation \( \log x^2 = (\log x)^2 \):- For \( x = 1 \), left side: \( \log 1^2 = 0 \), right side: \( (\log 1)^2 = 0 \).- For \( x = 100 \), left side: \( \log 100^2 = 4 \), right side: \( (\log 100)^2 = 4 \).Both satisfy the original equation.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental type of algebraic equation that can be expressed in the standard form: \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents variables. When solving quadratic equations, two main methods are typically used: factoring and utilizing the quadratic formula.
For equation \( y^2 - 2y = 0 \), factoring is chosen because it is the simplest approach. To factor a quadratic, find two numbers that multiply to the constant term \( c \) and add up to the coefficient \( b \). In this case, the equation factors as:

• \( y(y - 2) = 0 \)

This provides solutions\( y = 0 \) and \( y = 2 \), which are called the "roots" of the equation. These solutions are essential in determining the variable \( x \) in the original logarithmic equation.

Properties of Logarithms

Logarithms are used to deal with exponents conveniently. The properties of logarithms convert products into sums, ratios into differences, and powers into products.

• The Power Rule: This rule states \( \log a^b = b \cdot \log a \). It is crucial for transforming logarithmic equations and allows us to rewrite expressions for simplicity.
• The Product Rule: Expresses \( \log(ab) = \log a + \log b \), useful for expanding logarithmic expressions.
• The Quotient Rule: \( \log \left(\frac{a}{b}\right) = \log a - \log b \), simplifies division inside a logarithm.

In the given exercise, the Power Rule helps rewrite \( \log x^2 \) into \( 2 \cdot \log x \). This simplification allows the equation to take a form that is easier to solve, like an algebraic equation.

Analytic Methods

Analytic methods are systematic approaches to solving equations by manipulating them algebraically and logically. This can include substitution, factoring, and verifying solutions.
In the exercise, the substitution method simplifies the complex logarithmic equation into a manageable algebraic form. By letting \( y = \log x \), the equation \( \log x^2 = (\log x)^2 \) becomes a quadratic: \( y^2 - 2y = 0 \).

• Introduce a New Variable: By substituting \( \log x = y \), the equation is simplified.
• Rearrange and Factor: The equation is rearranged into a standard quadratic form and factored.
• Substitute Back: Solutions found in terms of \( y \) are substituted back into \( \log x = y \) to find \( x \).

This method ensures a concise and clear path, helping students grasp both logarithmic manipulation and quadratic solving systematically.