Problem 23 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \frac{1}{3} \log_{5} (x^2+1) \).

Step 1: Express the Root as an Exponent

Recall that a cube root can be rewritten as an exponent of one-third. Therefore, we can rewrite the expression \( \sqrt[3]{x^2+1} \) as \( (x^2 + 1)^{1/3} \). We want to expand \( \log_{5} (x^{2}+1)^{1/3} \).

Step 2: Apply the Power Rule of Logarithms

The Power Rule of Logarithms states that \( \log_b (a^n) = n \cdot \log_b a \). Applying this rule, we have that \( \log_{5} ((x^2 + 1)^{1/3}) = \frac{1}{3} \log_{5} (x^2+1) \).

Step 3: Verify Final Expression

The expression \( \frac{1}{3} \log_{5} (x^2+1) \) is the expanded form of the original logarithm according to the Laws of Logarithms. No further expansion is possible unless and until more known information is provided about \( x^2 + 1 \).

Key Concepts

Logarithmic Expansion

Expanding logarithmic expressions means to break them down into simpler components. We do this using the laws of logarithms. These rules help transform a complex logarithm into parts that are easier to manage. This makes solving equations or simplifying expressions possible.

In the given exercise, we see an example of this expansion process. When we start with an expression like \( \log_5(\sqrt[3]{x^2+1}) \), our goal is to rewrite it. We want it to contain no roots or powers within the logarithm. We do this by using our knowledge of exponents and logarithms.

The fundamental strategy here is substituting the root with an exponent. This paves the way to use more algebraic rules that will further simplify the expression.

Power Rule of Logarithms

The Power Rule of Logarithms is essential for handling expressions with exponents inside logarithms. It states: if you have \( \log_b(a^n) \), it can be rewritten to \( n \cdot \log_b(a) \).

This rule significantly simplifies calculations. When dealing with the original exercise, after expressing \( \sqrt[3]{x^2+1} \) as \( (x^2 + 1)^{1/3} \), the Power Rule allows us to pull the exponent \( \frac{1}{3} \) in front of the logarithm:
- \( \log_5((x^2+1)^{1/3}) = \frac{1}{3} \cdot \log_5(x^2+1) \)

Why is this so useful? It transforms a somewhat complex expression into a form where arithmetic operations become much more straightforward. This rule is one of the primary tools used to expand logarithmic functions effectively.

Properties of Exponents

Understanding the properties of exponents is crucial for manipulating expressions inside a logarithm. Exponents are a compact way of representing repeated multiplication, and they play a vital role in simplifying expressions.

One key property is the idea that any root can be considered a fractional power: \( \sqrt[3]{x} = x^{1/3} \). This property allows us to transform roots into exponents, which is necessary for applying the Power Rule of Logarithms.

• For example, the cube root \( \sqrt[3]{x^2+1} \) becomes \( (x^2+1)^{1/3} \).
• With exponents, we can also distribute multiplication over addition, though not quite as straightforwardly as multiplication, indicating the need for transformation and use of logarithms.

These properties make it possible to tackle more complex operations by breaking them down into simpler parts, which is especially handy when working with logarithms.