Problem 35 - Algebra A Combined Function

Question

Find the value of each logarithmic expression. $$ \log _{1 / 2} 2 $$

Step-by-Step Solution

Answer

The value of \( \log _{1 / 2} 2 \) is \( -1 \).

Step 1: Understand Logarithm Definition

The logarithm \( \log_{b}(a) = x \) is the exponent \( x \) to which the base \( b \) must be raised to produce the number \( a \). In this case, we need to find \( x \) such that \( \left( \frac{1}{2} \right)^x = 2 \).

Step 2: Rewrite Exponential Equation

Rewrite \( \left( \frac{1}{2} \right)^x = 2 \) using properties of exponents.\( \left( \frac{1}{2} \right)^x \) can be rewritten as \( 2^{-x} \), so the equation becomes \( 2^{-x} = 2 \).

Step 3: Equate Exponents

Since the bases are the same (both bases are \( 2 \)), equate the exponents to solve for \( x \): \[ -x = 1 \].

Step 4: Solve for \( x \)

Solve the equation \( -x = 1 \) for \( x \) by multiplying both sides by \( -1 \). \[ x = -1 \].

Key Concepts

Understanding Exponents

Exponents are a mathematical way to express repeated multiplication of a number by itself. For example, in the expression \(2^3\), 2 is the base, and 3 is the exponent, indicating that 2 is multiplied by itself three times, resulting in 8. Exponents can also be fractions or negative numbers, which add interesting dimensions to their behavior.

• Fractional exponents represent roots. For example, \(x^{1/2}\) is the square root of \(x\).
• Negative exponents denote reciprocal values. For instance, \(x^{-1}\) is the same as \(1/x\).

Understanding how to manipulate exponents is essential for solving equations, especially when they involve logarithms or other exponential functions. Remember, an expression like \((\frac{1}{2})^x\) is just an exponentiation of \(1/2\) by \(x\). This understanding helps us to transform and simplify expressions for easier computation.

The Nature of Logarithmic Expressions

Logarithms are the inverse operations of exponents, answering the question: "To what exponent must the base be raised to produce a certain number?" In mathematical terms, the logarithm \(\log_b(a) = x\) means that \(b^x = a\). For example, \(\log_{2}(8) = 3\) because \(2^3 = 8\).In the given exercise, we are asked to find \(\log_{1/2}(2)\). This logarithmic expression is asking us to determine what power the base \(1/2\) must be raised to in order to result in 2.

• Understanding the context of a logarithmic expression is crucial for translating it into a solvable equation.
• Logarithmic expressions can simplify complex exponential equations.

Once you recognize that \(\log_{1/2}(2)\) asks for the exponent that turns \(1/2\) into 2, you can proceed with transformations and solutions.

Solving Exponential Equations

To solve exponential equations, such as \((1/2)^x = 2\), the goal is to isolate the variable by equating powers. Firstly, rewrite the equation using properties of exponents. Knowing that \((1/2)^x\) can be expressed as \(2^{-x}\), and thus the equation \(2^{-x} = 2\) emerges.

• With the bases equal, equate the exponents: \(-x = 1\).
• Solving gives \(x = -1\) by multiplying both sides by -1.

This technique revolves around comparing the powers when the bases are consistent. Being confident with conversions and manipulations allows a seamless transition from one form to another, leading to straightforward solutions.