Problem 24 - Precalculus

Question

Solve the equation analytically. $$ \ln \left(x^{2}\right)=(\ln (x))^{2} $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = 1 \) and \( x = e^2 \).

Step 1: Simplification of the Equation

Start by recognizing that the left-hand side of the equation is the natural logarithm of a power. Using logarithmic identities, we can simplify: \( \ln(x^2) = 2\ln(x) \). Substituting this into the original equation gives us: \( 2\ln(x) = (\ln(x))^2 \).

Step 2: Rearrange the Equation

Rearrange the equation to form a standard quadratic expression: \( (\ln(x))^2 - 2\ln(x) = 0 \).

Step 3: Factor the Quadratic Expression

Factor the equation \( (\ln(x))^2 - 2\ln(x) = 0 \) by taking out a common factor: \( \ln(x)(\ln(x) - 2) = 0 \).

Step 4: Solve Each Factor

Solve each factor separately: - For \( \ln(x) = 0 \), raise \( e \) to both sides to find \( x = e^0 = 1 \). - For \( \ln(x) - 2 = 0 \), solve to find \( \ln(x) = 2 \). Thus, \( x = e^2 \).

Step 5: Verify Solutions

Check each potential solution in the original equation: - For \( x = 1 \), \( \ln(1^2) = (\ln(1))^2 = 0 \), which holds true. - For \( x = e^2 \), \( \ln((e^2)^2) = (\ln(e^2))^2 = 4, \) which also holds true.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra. These equations take the form of a polynomial equation of degree 2, generally written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The solutions to these equations are known as the roots or zeros of the polynomial, and they can be real or complex numbers.

Quadratic equations are solved using several methods such as:

• Factoring: Rewriting the equation as a product of binomials.
• Completing the square: Rearranging the equation so that one side becomes a perfect square trinomial.
• The quadratic formula: Applying \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a \), \( b \), and \( c \) are the coefficients of the equation.

Understanding the basic form of quadratic equations helps in simplifying problems like the one in the exercise by allowing us to arrange expressions into a format we can solve with these methods.

Factoring

Factoring is a crucial algebraic skill that involves breaking down an equation into simpler components, typically products of factors. In the context of quadratic equations, factoring often refers to expressing a quadratic polynomial as the product of two binomial expressions.

In our exercise, we saw the equation \((\ln(x))^2 - 2\ln(x) = 0\). This expression is already set to be factored since a common term, \(\ln(x)\), appears twice. By taking \(\ln(x)\) as a factor, we can rewrite the equation as:

• \(\ln(x)(\ln(x) - 2) = 0\)

This procedure simplifies the solving process by producing two smaller equations that can be solved independently. Factoring is an indispensable tool since it can convert a seemingly complex equation into more manageable sub-equations.

Logarithmic Identities

Logarithmic identities are rules that govern the transformation and simplification of logarithmic expressions. They are particularly useful for converting complex logarithmic equations into simpler forms that are easier to manipulate and solve.

Some important logarithmic identities include:

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

In the given exercise, we applied the power rule to simplify \( \ln(x^2) \) to \( 2\ln(x) \). These identities are powerful because they allow the transformation of logarithmic expressions into linear or quadratic forms, which can be more directly solved.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a logarithmic function with base \( e \), where \( e \approx 2.71828 \). It is a fundamental constant in mathematics found in various natural contexts such as growth processes, physics, and more.

In solving equations like our exercise, the natural logarithm is prevalent due to its properties that allow the conversion of exponential equations into linear ones. The natural logarithm function has several important properties:

• \( \ln(e) = 1 \)
• \( \ln(1) = 0 \)
• Inverse property: \( e^{(\ln(x))} = x \)

In the solution process, we used the properties of natural logarithms to evaluate solutions such as \( x = 1 \) and \( x = e^2 \). By understanding these fundamental concepts, the natural logarithm becomes a reliable tool in handling exponential growth and decay problems, as well as simplifying logarithmic and exponential equations.