Problem 23 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{2} \sqrt[4]{8}$$

Step-by-Step Solution

Answer

The exact value is \(3/4\).

Step 1: Rewrite Fourth Root as Fractional Power

The fourth root of 8 can be rewritten as a power with a fractional exponent. This would give \(\log_2 (8^{1/4})\).

Step 2: Implement Property of Logarithms

By the power rule for logarithms, which states \(\log_b{a^k} = k\log_b{a}\), this expression can be written as \(1/4 \times \log_2 8\).

Step 3: Evaluate Logarithm

The log base 2 of 8 is 3 because \(2^3 = 8\). Thus, this evaluates to \(1/4 \times 3\).

Step 4: Simplify Fraction

Simplify \(1/4 \times 3\) to get the final answer.

Key Concepts

Properties of Logarithms

Logarithms have several useful properties that make them a powerful tool in mathematics. These properties allow us to simplify and evaluate logarithmic expressions more easily. One of the key properties is the product property, which states that the logarithm of a product is the sum of the logarithms of the factors:

• \( \log_b{(MN)} = \log_b{M} + \log_b{N} \)

The quotient property allows us to express the logarithm of a quotient as the difference of two logarithms:

• \( \log_b{ \left( \frac{M}{N} \right) } = \log_b{M} - \log_b{N} \)

Finally, the power property is particularly useful for this exercise, as it helps handle expressions raised to exponents. This property is expressed as follows:

• \( \log_b{a^k} = k\log_b{a} \)

Understanding these properties provides the foundation for solving logarithmic equations and expressions without a calculator.

Power Rule for Logarithms

The power rule for logarithms is a convenient shortcut that helps handle expressions where a term is raised to a power. According to this rule, when you have a logarithm of a power, such as \( \log_b{a^k} \), it can be simplified by bringing the exponent \( k \) in front of the logarithm:

• \( \log_b{a^k} = k \log_b{a} \)

This rule simplifies the calculation by turning a potentially complex expression into a more straightforward multiplication.
In the exercise given, we used this rule to rewrite the expression \( \log_2{(8^{1/4})} \) as \( \frac{1}{4} \log_2{8} \). By leveraging the power rule, we reduce the complexity of solving for the logarithm, allowing us to efficiently evaluate expressions without any advanced calculation tools.

Fractional Exponent

Fractional exponents are a way of expressing roots as powers, which can be crucial in simplifying logarithmic expressions. For instance, the fourth root of a number \( N \) can be written with a fractional exponent as \( N^{1/4} \).
This form is particularly useful in logarithmic calculations due to logarithmic properties.
By expressing roots as fractional exponents, we can utilize the power property of logarithms, making expressions easier to evaluate. In the context of the exercise, we took the fourth root of 8 and wrote it as \( 8^{1/4} \).
This step allowed us to apply the power rule for logarithms directly, facilitating the simplification of the entire expression. Understanding how to work with fractional exponents can greatly ease the manipulation and computation of advanced mathematical problems.

Logarithmic Expressions

Logarithmic expressions can often appear daunting, but they usually break down into simpler calculations using the right properties and rules. These expressions involve the operation of taking the logarithm of a number and can be expressed in different bases.
A critical aspect is choosing the correct properties or rules to simplify or evaluate them. In this exercise involving \( \log_2 \sqrt[4]{8} \), we expressed the root as \( 8^{1/4} \), applied the power rule, and found \( \log_2 8 \) equals 3 since \( 2^3 = 8 \).
Logarithmic expressions harness the power of inverse exponentiation, allowing us to solve equations involving exponential growth or decay. By mastering the properties and rules of logarithms, one can tackle complex logarithmic operations smoothly and efficiently, turning seemingly intricate problems into straightforward tasks.