Problem 15 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b} x^{3} $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \( 3 \cdot \log_{b}{x} \).

Step 1: Identify the Expression

Firstly, note that the given logarithmic expression is \( \log _{b} {x^{3}} \)

Step 2: Apply Power Rule

Applying the power rule of logarithms to the expression, it expands to \( 3 \cdot \log_{b}{x} \).

Key Concepts

Logarithmic Expressions

Logarithmic expressions are a way to express exponentiation in a different form, involving a base and an exponent as part of the logarithm function. In their simplest form, they are written as \( \log_b{a} \) which asks the question, 'To what power should we raise the base \( b \) to obtain \( a \)?' Understanding logarithmic expressions is crucial in various fields of mathematics, especially when dealing with exponential growth, decay, and complex calculations in algebra and calculus.

When working with logarithmic expressions, there are specific rules that we rely on to simplify or manipulate the expression. These include the product rule, quotient rule, and power rule. Each rule has a specific function, for example, turning the multiplication of numbers into the addition of their corresponding logarithms, or in the case of the power rule, reducing the power of a logarithm to a simple multiplication. It's akin to breaking down a complex mechanism into simpler, manageable parts, thereby making tough calculations much easier to handle without the use of a calculator.

Power Rule of Logarithms

The power rule of logarithms is a handy tool that helps us simplify logarithmic expressions, especially when the argument of the logarithm is raised to a power. This rule states that \( \log_b{a^n} = n \cdot \log_b{a} \), which means that if you have a logarithm with an exponent, you can 'bring down' the exponent to the front of the logarithm and multiply it.

This is incredibly useful when expanding logarithmic expressions or when aiming to make expressions more amenable to calculation without a calculator. For instance, dealing with \( \log _{b} x^{3} \) can seem difficult at first, but by applying the power rule, it becomes \( 3 \cdot \log_{b}{x} \) — much more straightforward! When utilizing the power rule, it's important to remember that the base of the logarithm remains the same; only the exponent is affected by this rule.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves using the properties of logarithms to rewrite a single logarithmic expression as several simpler terms. This practice is particularly useful when trying to solve logarithmic equations or when simplifying expressions for easier computation. The process of expansion usually involves applying the power, product, and quotient rules.

Take for example the expression \( \log _{b} x^{3} \). To expand this expression, we look for exponents, products, or quotients within the logarithm to apply the corresponding rules. Here, we identify an exponent, '3', and apply the power rule of logarithms which allows us to move the exponent to the front. Consequently, we expand the expression to \( 3 \cdot \log_{b}{x} \).

The goal of expansion is to make the expression as transparent and easy to work with as possible. It's a bit like putting together a puzzle where each piece must fit perfectly to reveal the bigger picture. When a student understands how to expand logarithmic expressions correctly, they gain the power to tackle more complex logarithmic equations with confidence.