Problem 26 - Intermediate Algebra

Question

For Problems \(21-40\), evaluate each expression. $$ \log _{4} 256 $$

Step-by-Step Solution

Answer

The logarithm is 4.

Step 1: Understand the Problem

The expression \( \log_{4} 256 \) asks us to determine the exponent to which the base 4 must be raised to produce 256.

Step 2: Apply the Definition of Logarithms

Based on the definition of logarithms, \( \log_{b} a = c \) means that \( b^{c} = a \). Here, we need to find \( x \) such that \( 4^{x} = 256 \).

Step 3: Express the Number as a Power of the Base

Recognize that 256 is a power of 4. We can express 256 as \( 4^4 \) because \( 4 \times 4 \times 4 \times 4 = 256 \).

Step 4: Solve for the Exponent

Since 256 can be written as \( 4^4 \), it follows that \( \log_{4} 256 = 4 \) because 4 is the exponent needed to raise 4 to get 256.

Key Concepts

Exponents

Let's start with understanding what exponents are. Exponents are a shorthand method to represent repeated multiplication of the same number. For example, in the expression \( 3^4 \), 3 is the base that is multiplied by itself four times. This can be expanded as \( 3 \times 3 \times 3 \times 3 = 81 \). The number above the base, 4, is the exponent, and it tells us how many times to use the base as a factor. In mathematical operations, exponents are quite powerful because they simplify calculations and expressions. Instead of writing out a long sequence of multiplications, we can convey the same idea more concisely with exponent notation.

• Exponents apply to any number, positive or negative.
• The exponent can also be zero or negative, both of which have special rules and meanings in mathematics.

Logarithms are essentially the opposite of exponentiation, helping us to find the exponent in expressions where the result is already known.

Powers of Numbers

Powers of numbers are integral to understanding exponents. When we talk about powers, we usually refer to expressions like \( b^n \), where \( b \) is the base and \( n \) is the exponent. Powers help us determine how many times a number multiplies itself. This forms the backbone of exponential growth and decay in mathematics.Consider the expression \( 2^3 \). Here, 2 is the base, and 3 is the exponent, meaning \( 2 \times 2 \times 2 = 8 \). Regardless of the complexity of the base—whether it's a fraction, a decimal, or a whole number—the essence of the operation remains the same: repeated multiplication.

• Understanding powers helps in simplifying complex calculations.
• Powers are used in various fields, including science, engineering, and finance, where exponential growth or decay is modeled.

Recognizing a number as a power of another can also aid in solving logarithm problems, like determining \( \log_{4} 256 \) by knowing that 256 is \( 4^4 \).

Base and Exponent Relationship

The relationship between a base and its exponent forms the core of many mathematical expressions. The base is the number being multiplied, while the exponent indicates the number of times the multiplication occurs. Each part has a distinct role, and understanding this relationship is crucial when dealing with logarithms and exponential equations. For example, in the equation \( 5^3 = 125 \), the base 5 is what gets multiplied three times as per the exponent, resulting in 125. In the realm of logarithms, like \( \log_{b} a = c \), it is this relationship that facilitates the determination of the unknown exponent \( c \), such that \( b^c = a \).

• The base must be a positive number except in some advanced mathematical contexts.
• As the exponent increases, the resultant value of the power grows exponentially fast.

In the example from the original problem, understanding that 256 can be expressed as a power of 4 (\( 4^4 \)), allows us to solve \( \log_{4} 256 \) easily, identifying the exponent as 4.