Problem 29 - Precalculus Mathematics for Calculus

Question

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: Rewrite the Radical as an Exponent

The expression \( \log_{5} \sqrt[3]{x^{2}+1} \) contains a cube root. A cube root can be rewritten as an exponent of \( \frac{1}{3} \). So, express \( \sqrt[3]{x^{2}+1} \) as \( (x^{2}+1)^{\frac{1}{3}} \). This gives us \( \log_{5}(x^{2}+1)^{\frac{1}{3}} \).

Step 2: Use the Power Rule of Logarithms

The Power Rule of Logarithms states that \( \log_{b} (a^{n}) = n \cdot \log_{b} (a) \). Apply this rule to the rewritten expression: \( \log_{5}((x^{2}+1)^{\frac{1}{3}}) = \frac{1}{3} \cdot \log_{5}(x^{2}+1) \).

Key Concepts

Logarithmic Expansion

Logarithmic expansion is a method used to simplify complex logarithmic expressions. This technique allows us to express a single logarithm as a combination of several simpler logs, making the evaluation and understanding more manageable.
By using the laws of logarithms, such as the Power Rule, Product Rule, and Quotient Rule, we can break down expressions into their components. In the context of our exercise, we focus on using the Power Rule to efficiently and accurately expand logarithmic expressions.
The transformation process starts with the given complex expression and continues through logical steps, making it easier to understand and solve mathematical problems efficiently.

Power Rule of Logarithms

The Power Rule of Logarithms is a fundamental principle that helps to simplify logarithmic expressions involving exponents. The rule states that for any logarithmic function of the form \( \log_{b}(a^n) \), it equates to \( n \cdot \log_{b}(a) \).
This means if you have an exponent within the logarithm, you can bring the exponent in front as a multiplier. This step significantly simplifies the expression, transforming it from a power to a simple multiple.
In our exercise, the cube root of \( x^2 + 1 \) is initially expressed as an exponent of \( \frac{1}{3} \). Applying the Power Rule, we move \( \frac{1}{3} \) in front, making the expression easier to manage with subsequent steps.

Cube Root as Exponent

Understanding how roots can be represented as exponents is a crucial part of simplifying complex expressions in mathematics. Every root has an equivalent expression as a fractional exponent. For instance, the cube root of a number is the same as raising that number to the power of \( \frac{1}{3} \).
This relationship allows us to work with expressions in a more straightforward manner, especially when they involve logarithms. When you convert roots to exponents, it becomes easier to apply logarithmic rules like the Power Rule.
In our exercise, the cube root \( \sqrt[3]{x^2+1} \) is rewritten as \( (x^2+1)^{\frac{1}{3}} \), setting up the expression for further simplification using logarithmic rules.