Problem 15 - Precalculus with Calculus Previews

Question

Solve the given exponential equation. $$ \frac{1}{3}=\left(2^{|x|-2}-1\right)^{-1} $$

Step-by-Step Solution

Answer

The solutions are \( x = 4 \) and \( x = -4 \).

Step 1: Understanding the Equation

The given equation is \( \frac{1}{3} = \left(2^{|x|-2} - 1\right)^{-1} \). This means that \( 1/3 \) is the reciprocal of \( 2^{|x|-2} - 1 \).

Step 2: Taking the Reciprocal

To solve for \( |x| \), first take the reciprocal of both sides. This gives us \( 3 = 2^{|x|-2} - 1 \).

Step 3: Isolating the Exponential Term

Add 1 to both sides to isolate the exponential expression. We get \( 2^{|x|-2} = 4 \).

Step 4: Equating Bases

Since \( 4 = 2^2 \), rewrite \( 4 \) as \( 2^2 \). Therefore, \( 2^{|x|-2} = 2^2 \).

Step 5: Solving the Exponent

Since the bases are the same, equate the exponents: \( |x|-2 = 2 \).

Step 6: Solving for |x|

Add 2 to both sides to solve for \( |x| \). Thus, \( |x| = 4 \).

Step 7: Determine Values of x

Since \( |x| = 4 \), \( x \) can be either 4 or -4.

Key Concepts

Understanding Absolute Value

Absolute value refers to the distance between a number and zero on the number line, without considering which direction from zero the number lies. The absolute value of a number is always non-negative. For instance, both 4 and -4 are four steps away from zero, so their absolute values are equal, expressed as \(|4| = 4\) and \(|-4| = 4\).
In the context of equations, absolute values are often used to indicate that a solution could have either a positive or a negative form. For example, if you have an equation like \(|x| = 4\), it means \(x\) can be either 4 or -4. Recognizing absolute value helps anticipate multiple solutions during equation solving.

• Always remember that the absolute value transforms any negative sign into a positive one.
• Solutions to equations with absolute values must be checked to satisfy the original equation.

Absolute values are vital because they help us appreciate the magnitude of a number, much like considering the size of a shadow without thinking of the light's direction.

Mastering Reciprocals

A reciprocal is essentially flipping the numerator and the denominator of a fraction. Mathematically, the reciprocal of a number \(a\) is \(\frac{1}{a}\). Reciprocals are important in manipulating equations and solving problems involving ratios and proportions.
To illustrate, the reciprocal of \(\frac{1}{3}\) is 3, because when you multiply them, you get 1: \(\frac{1}{3} \times 3 = 1\). This concept is particularly useful when dealing with equations where a term is raised as a reciprocal, allowing us to transform the equation step by step without altering its equality.

• Reciprocals help simplify equations, especially when terms are part of a fraction.
• Remember that the reciprocal of 0 is undefined since division by 0 is not permissible.

Understanding reciprocals is fundamental, as they exhibit symmetry and balance in equations, offering a pathway to simplify complex mathematical expressions.

Exploring Exponents

Exponents are a way to express repeated multiplication of a number by itself. For instance, \(2^3\) denotes \(2 \times 2 \times 2\). Exponents are crucial in exponential equations, helping to describe rates of growth, decay, and complex mathematical ideas succinctly.
In exponential equations, different bases and powers allow us to simplify or expand expressions rapidly. In our example \(2^{|x|-2} = 4\), rewriting 4 as an exponent of 2, i.e., \(4 = 2^2\), allows matching the bases to solve the equation simply by equating their exponents.

• Learning exponent rules streamlines solving equations, such as multiplying powers and powers of powers.
• Remember the Zero Exponent Rule: Any non-zero number raised to the power of 0 equals 1.

Understanding exponents and their properties provides a mathematical shortcut, transforming challenging problems into manageable solutions.

Solving Exponential Equations

Solving exponential equations involves finding the unknown value in expressions where variables appear as exponents. It is a methodical process often requiring the transformation of the base or manipulation of the equation to isolate the variable.
The first step often involves recognizing forms that can be rewritten with the same base, which simplifies comparing exponents. As seen in the problem \(3 = 2^{|x|-2} - 1 \), taking the reciprocal is a strategy that flips the equation, facilitating simpler manipulation by isolating the exponential part.

• Aim to express all terms with the same base to direct solve exponents.
• Check all potential solutions in the original equation to confirm they fit all criteria.

Mastery of solving exponential equations builds essential problem-solving skills and expands understanding in numerous fields, including biology for growth curves and finance for compound interest.