Problem 29 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log _{2}\left(\frac{x\left(x^{2}+1\right)}{\sqrt{x^{2}-1}}\right) $$

Step-by-Step Solution

Answer

\( \log_2 x + \log_2(x^2+1) - \frac{1}{2} \log_2(x^2-1) \)

Step 1: Apply Logarithm of a Fraction

Use the property \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \) to split the given logarithmic expression: \[ \log_2 \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) = \log_2(x(x^2+1)) - \log_2(\sqrt{x^2-1}) \]

Step 2: Expand the Numerator using Logarithm of a Product

Apply the property \( \log_b(MN) = \log_b M + \log_b N \) to the term \( \log_2(x(x^2+1)) \): \[ \log_2(x(x^2+1)) = \log_2 x + \log_2(x^2+1) \]

Step 3: Simplify the Logarithm of a Root

Use the property \( \log_b(N^{1/n}) = \frac{1}{n} \log_b N \) to simplify \( \log_2(\sqrt{x^2-1}) \). Since \( \sqrt{x^2-1} = (x^2-1)^{1/2} \), apply the property: \[ \log_2(\sqrt{x^2-1}) = \frac{1}{2} \log_2(x^2-1) \]

Step 4: Combine All Parts

Substitute the expanded components back into the logarithmic expression:\[ \log_2 \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) = \log_2 x + \log_2(x^2+1) - \frac{1}{2} \log_2(x^2-1) \] This result gives the expanded form using the Laws of Logarithms.

Key Concepts

Logarithm of a Fraction

The logarithm of a fraction allows us to separate the expression inside a logarithm when dealing with division. In general, the rule is expressed as:

• \( \log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N \)

This means that the logarithm of a fraction can be expanded into the difference of two logarithms.
For instance, given an expression like \( \log_2 \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) \), we can use this rule to split it into two parts:
\[ \log_2(x(x^2+1)) - \log_2(\sqrt{x^2-1}) \]
This is useful because it simplifies the overall expression, breaking it down into more manageable pieces.

Logarithm of a Product

When dealing with the product of two numbers inside a logarithm, we utilize the logarithm of a product rule. The general property is:

• \( \log_b(MN) = \log_b M + \log_b N \)

This property allows us to convert a multiplication within a logarithm into an addition of logarithms.
For example, with the expression \( \log_2(x(x^2+1)) \), we can expand it using this property:
\[ \log_2 x + \log_2(x^2+1) \]
By doing so, we make it easier to handle each part separately, further breaking down complex expressions into simpler components.

Logarithm of a Root

When you have a logarithm of a root, you can simplify it by using the following logarithmic property:

• \( \log_b(N^{1/n}) = \frac{1}{n} \log_b N \)

This tells us that a root in the logarithm can be expressed as a multiplication outside the logarithm.
For instance, take \( \log_2(\sqrt{x^2-1}) \). Recognizing \( \sqrt{x^2-1} \) as \( (x^2-1)^{1/2} \), we apply the property:
\[ \frac{1}{2} \log_2(x^2-1) \]
This shifts the root operation into a fraction, simplifying the expression significantly.
This technique is especially beneficial when simplifying expressions that involve square roots or other roots.

Logarithmic Expansion

Logarithmic expansion refers to the process of breaking down complex logarithmic expressions into simpler, more computable parts using different logarithmic properties. Here's how it works:
We start with a complex expression, such as \( \log_2 \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) \), and use various logarithmic rules to expand it. By employing the laws like the logarithm of a fraction, product, and root, we derive a more manageable form:
\[ \log_2 \left( \frac{x(x^2+1)}{\sqrt{x^2-1}} \right) = \log_2 x + \log_2(x^2+1) - \frac{1}{2} \log_2(x^2-1) \]
This expansion process is crucial in simplifying complex expressions for further mathematical manipulation or evaluation.
By mastering logarithmic expansion, you effectively decode large or intimidating expressions, making them easier to work with in various mathematical contexts.