Problem 569 - Precalculus

Question

Rewrite \(\log _{8}\left(a^{\frac{1}{b}}\right)\) as a product.

Step-by-Step Solution

Answer

\( \log_8\left(a^{\frac{1}{b}}\right) = \frac{1}{b} \cdot \log_8(a) \).

Step 1: Use the Power Rule of Logarithms

The power rule of logarithms states that \( \log_b(x^y) = y \cdot \log_b(x) \). In this expression, \( x = a \) and \( y = \frac{1}{b} \). Apply the rule: \[ \log_8\left(a^{\frac{1}{b}}\right) = \frac{1}{b} \cdot \log_8(a) \].

Step 2: Express as a Product

The expression \( \frac{1}{b} \cdot \log_8(a) \) is already written as a product. The coefficient \( \frac{1}{b} \) is being multiplied by \( \log_8(a) \). Thus, the expression is \( \frac{1}{b} \cdot (\text{some function}) \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental property that helps to simplify logarithmic expressions when dealing with exponents. When you see a logarithm in the form \( \log_b(x^y) \), you can apply the power rule to bring down the exponent as a coefficient.
This rule states that \( \log_b(x^y) = y \cdot \log_b(x) \). In other words, any exponent inside the logarithm can be placed in front as a multiplication factor or coefficient.
This property is particularly useful for simplifying complex logarithmic expressions or when we need to solve equations involving logarithms. For example, in the original exercise, \( a^{\frac{1}{b}} \) had the exponent \( \frac{1}{b} \), which was brought in front of the logarithm to transform the expression.

Logarithmic Expressions

A logarithmic expression involves a logarithm, which is the inverse operation to exponentiation. Such expressions can vary from simple to complex forms, depending on various factors like the base and argument. Understanding how to manipulate these expressions allows you to work with them more effectively.
For example, expressions like \( \log_8(a) \) tell you the power to which the base 8 must be raised to yield the number \( a \). Simplifying these expressions often involves using logarithmic properties, such as the power rule or the product rule, to make them more manageable.
In various fields of mathematics and science, accurately rewriting or simplifying logarithmic expressions is crucial for solving numerous types of problems.

Logarithm Multiplication

Logarithm multiplication involves using coefficients that multiply the logarithmic function itself. In our example, the expression becomes \( \frac{1}{b} \cdot \log_8(a) \) after applying the power rule.
This is essentially a multiplication of \( \log_8(a) \) by \( \frac{1}{b} \), where \( \frac{1}{b} \) acts as a scaling factor.
Multiplication like this does not change the fundamental operation of the logarithm but modifies its value based on the coefficient. Understanding multiplication in logarithmic expressions is vital when converting into simpler forms or evaluating them, especially when solving real-world problems where precise calculations are necessary.