Problem 29 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{5} \sqrt[3]{x^{2}+1} $$

Step-by-Step Solution

Answer

\(\frac{1}{3} \cdot \log_5 (x^2 + 1)\)

Step 1: Convert Radical to Exponent Form

The given expression involves a cube root, which can be rewritten using exponents. The cube root of \( x^2 + 1 \) is \( (x^2 + 1)^{1/3} \). Hence, the expression becomes \( \log_5 ((x^2 + 1)^{1/3}) \).

Step 2: Apply the Power Rule of Logarithms

According to the power rule of logarithms, \( \log_b (a^c) = c \cdot \log_b a \). We identify \( a = x^2 + 1 \) and \( c = \frac{1}{3} \). Thus, we apply the rule as follows: \( \log_5 ((x^2 + 1)^{1/3}) = \frac{1}{3} \cdot \log_5 (x^2 + 1) \).

Step 3: Expression in Expanded Form

The expanded form of the original expression \( \log_5 \sqrt[3]{x^2 + 1} \) is \( \frac{1}{3} \cdot \log_5 (x^2 + 1) \). Therefore, you have successfully used the laws of logarithms to expand the expression.

Key Concepts

Understanding Logarithmic Expansion

Logarithmic expansion is a technique used to simplify complex logarithmic expressions into more manageable forms. This process involves using various logarithmic laws, such as the power, product, and quotient rules, to break down expressions. In our example, we expand the logarithmic expression \( \log_5 \sqrt[3]{x^2 + 1} \) by realizing it involves a root, which can be rewritten in a simpler exponent form.

By understanding and applying these laws, complex expressions become easier to work with, both conceptually and computationally. It's essential for problem-solving in mathematics, especially in calculus and algebra.

• Recognize parts of the expression that can be simplified.
• Apply the appropriate logarithmic laws.
• Rewrite the expression in its expanded form.

This makes problem-solving more intuitive and helps in performing further mathematical operations with ease.

Demystifying the Power Rule of Logarithms

The power rule of logarithms is a tool that lets us bring exponents out in front of a logarithm, making expressions simpler to handle. According to this rule, \( \log_b (a^c) = c \cdot \log_b a \). In our example, we have \( (x^2 + 1)^{1/3} \), where the exponent \( c \) is \( \frac{1}{3} \).

By applying the power rule, we can simplify the input to the logarithm to become \( \frac{1}{3} \cdot \log_5 (x^2 + 1) \), which is often easier to work with in calculations.

• Identify the exponent in the expression.
• Apply the power rule to simplify it.
• Check your results by comparing terms before and after.

This rule is a fundamental part of the logarithmic toolkit that helps break down expressions with exponents.

Exploring the Exponential Form

An integral part of working with logarithms involves understanding their relationship with exponents. The exponential form allows us to change roots and powers into expressions involving exponents. For instance, a cube root like \( \sqrt[3]{x^2 + 1} \) can be rewritten as \( (x^2 + 1)^{1/3} \). This conversion is crucial when applying logarithmic rules as it turns the root operation into a more familiar exponent form.

This is particularly useful because logarithmic operations are mostly defined for exponent-based expressions. When expressions are converted from radical to exponential form, we can easily apply the power rule or other logarithmic laws to simplify or solve problems.

• Recognize when an expression can be rewritten in exponential form.
• Use this form to apply logarithmic rules effectively.
• Practice converting between radical and exponential forms.

Understanding this relationship is fundamental in higher-level mathematics, enhancing problem-solving skills by allowing a broader application of logarithmic rules.