Question

Expand the logarithmic expression. (See Example 2.) $$\ln 3 x^4$$

Short answer

The expansion of the logarithmic expression \( \ln 3 x^4 \) is \( \ln 3 + 4 * \ln x \).

Step-by-step solution

Identify The Properties of Logarithms

For a given base b, \ln b(mn) = \ln b(m) + \ln b(n) is the product rule and \ln b(m^p) = p * \ln b(m) is the power rule.

Apply The Properties of Logarithms

Rewrite the given expression \( \ln 3 x^4 \) in accordance with the product rule: this becomes \( \ln 3 + \ln x^4 \).

Simplify with Power Rule

Now apply the power rule to the expression. This gives \( \ln 3 + 4 * \ln x \).

Key concepts

Properties of Logarithms

Logarithms have several useful properties that make it easier to work with complex expressions. Understanding these properties helps in simplifying and expanding logarithmic expressions. The properties include:

• Product Rule
• Power Rule
• Quotient Rule

In this context, we are focusing on the first two. The **Product Rule** is used when you have two numbers being multiplied inside the logarithm. The rule states that \( \ln(mn) = \ln(m) + \ln(n) \). It essentially transforms the product inside the log into a sum outside it.
Another property is the **Power Rule**, which applies to logarithms with expressions raised to a power. If you have a power inside the logarithm, this rule simplifies it by bringing the power out in front: \( \ln(m^p) = p \cdot \ln(m) \). This is particularly handy when dealing with powers inside logarithms.
By knowing and applying these properties, you can effectively break down logarithmic expressions into simpler parts.

Product Rule

The **Product Rule** for logarithms is a fundamental tool for breaking down expressions where numbers or variables are multiplied together inside a logarithm. Think of it as untangling a multiplication into an addition of simpler logs.
When you encounter an expression like \( \ln(3x^4) \), the Product Rule can be applied:

• Identify the product inside the logarithm, in this case, \( 3 \times x^4 \).
• Break it down using the rule: \( \ln(3x^4) = \ln(3) + \ln(x^4) \).

This step divides the expression into separate parts that can each be handled independently. It’s like breaking down a complex task into smaller, manageable pieces. Utilizing the Product Rule effectively splits the expression into simpler logarithmic parts, paving the way for easier application of further properties.

Power Rule

When faced with expressions involving exponents inside logarithms, the **Power Rule** becomes your best friend. The rule exploits the relationship between powers and logarithms by moving the exponent to the front of the log expression, making calculations straightforward.

• Consider an expression like \( \ln(x^4) \).
• The Power Rule states that \( \ln(x^4) = 4 \cdot \ln(x) \).

By doing this, the exponent "4" comes out front, and what remains is a simpler logarithmic term \( \ln(x) \). This transformation is particularly useful as it converts a more complex power into a simple multiplication, making it much easier to handle or calculate further.
Overall, the Power Rule allows you to streamline expressions, transforming them into a simpler form that is typically quicker to evaluate or combine with other log expressions. It’s a powerful technique to simplify terms with exponents, and when used in conjunction with other rules, leads to clear and concise results.