Problem 41 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$\log _{1 / 2}\left(\frac{1}{4}\right)$$

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Identify Logarithmic Base and Argument

The problem requires simplifying \( \log_{1/2}(1/4) \). Here, the base of the logarithm is \( \frac{1}{2} \), and the argument is \( \frac{1}{4} \).

Step 2: Understand Logarithm Definition

Recall that \( \log_b(a) = x \) means \( b^x = a \). Thus, we seek a power \( x \) such that \( \left(\frac{1}{2}\right)^x = \frac{1}{4} \).

Step 3: Express Argument in Terms of Base

Express \( \frac{1}{4} \) as a power of \( \frac{1}{2} \). Since \( \frac{1}{4} = \left(\frac{1}{2}\right)^2 \), we can write the equation as \( \left(\frac{1}{2}\right)^x = \left(\frac{1}{2}\right)^2 \).

Step 4: Solve for x

Since both sides of the equation have the same base, set the exponents equal: \( x = 2 \).

Step 5: Simplify the Expression

Therefore, the expression \( \log_{1/2}(1/4) \) simplifies to \( 2 \).

Key Concepts

Simplification

Simplification involves making expressions easier to handle and understand by reducing them to their simplest form. For logarithms, simplification often means transforming a complex logarithmic expression into a simpler numerical value or a more manageable expression.

• Identify the components of the expression such as the base and the argument.
• Use mathematical properties or known identities to simplify.
• Solve any resulting equations to isolate variables or constants.

In the exercise, the simplification process involves recognizing that the logarithmic expression \( \log_{1/2}(1/4) \) can be expressed as a simple integer. By breaking down the expression and using properties of exponents and logarithms, we discovered that \( x = 2 \). This shows how simplification can help transform a complex logarithmic problem into a straightforward answer.

Logarithmic Functions

Logarithmic functions play a crucial role in mathematics by serving as the inverse operations of exponential functions. If you have an exponential relationship, a logarithm allows you to solve for the unknown exponent.
For example, \( \log_b(a) = x \) indicates that \( b^x = a \). This means you are looking for the power \( x \) to which the base \( b \) must be raised to yield \( a \).

• Logarithms convert multiplication into addition, making them very useful for solving equations involving exponential growth or decay.
• Understanding the base and the argument is key in computing logarithms.
• Logarithms follow specific rules, like the Power Rule \( \log_b(a^n) = n \log_b(a) \) which helps with simplification.

In the given exercise, understanding that the base \( \frac{1}{2} \) and the argument \( \frac{1}{4} \) can be expressed with the same base helps us use the inverse property of exponents to solve the problem efficiently.

Exponents

Exponents are critical in the realm of mathematics, reflecting how many times a number, known as the base, is multiplied by itself. They simplify the representation of repeated multiplication.

• An exponent of 2, for instance, means multiplying the base by itself once, \( a^2 = a \times a \).
• Different exponent rules, like the Power Rule \( (b^m)^n = b^{mn} \), simplify complex calculations.
• Converting expressions using exponents into those with a common base can make solving equations straightforward.

The step-by-step solution hinges on transforming \( \frac{1}{4} \) into a power of \( \frac{1}{2} \), recognizing that \( \frac{1}{4} = \left(\frac{1}{2}\right)^2 \). This shows how powerful and versatile exponents are in simplifying and solving logarithmic expressions.