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Chapter 4: Problem 13

Simplify using logarithm properties to a single logarithm. $$ 2 \log (x)+3 \log (x+1) $$

Short Answer

Expert verified
\( \log(x^2(x+1)^3) \)

Step by step solution

01

Apply Power Rule

The power rule for logarithms states that \( a \log b = \log(b^a) \). Thus, apply the power rule to each term in the expression: - \( 2 \log (x) = \log(x^2) \)- \( 3 \log (x+1) = \log((x+1)^3) \)
02

Apply Product Rule

The product rule for logarithms states that \( \log a + \log b = \log(ab) \). Use this rule to combine the terms obtained from the power rule: \[ \log(x^2) + \log((x+1)^3) = \log(x^2(x+1)^3) \].
03

Simplify the Expression

Finally, combine the expressions inside the logarithm if possible from the product rule result:The single logarithm simplified form is: \( \log(x^2(x+1)^3) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Power Rule
The power rule is an essential logarithmic property used to simplify expressions where a logarithm is multiplied by a constant. It states:
  • \( a \log b = \log(b^a) \)
This means you can "move" the constant from in front of the log inside the log as an exponent. For example, in the expression \( 2 \log(x) \), the 2 becomes the exponent of \( x \), resulting in \( \log(x^2) \).
Similarly, \( 3 \log(x+1) \) becomes \( \log((x+1)^3) \). Moving coefficients as exponents simplifies the handling of sums and products in logarithms.
Mastering the Product Rule
The product rule is another key property in logarithms helpful for combining logs into a single expression. This rule is written as:
  • \( \log a + \log b = \log(ab) \)
This property allows you to convert the addition of two logarithms into the logarithm of a product. For example, if you have \( \log(x^2) + \log((x+1)^3) \), you can combine them using the product rule into \( \log(x^2(x+1)^3) \).
This is immensely useful when aiming to simplify expressions by reducing multiple logarithmic terms into one.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions involves using rules such as the power rule and product rule to condense multiple logs into a single log expression. The steps generally include:
  • Applying the Power Rule: Adjust coefficients as exponents inside the log.
  • Using the Product Rule: Combine addition of logarithms into a single logarithmic product.
In this particular exercise, applying these rules to \( 2 \log(x) + 3 \log(x+1) \) results in the single, simplified form: \( \log(x^2(x+1)^3) \). Simplifying in this manner reduces complexity and reveals underlying mathematical relationships.
Mathematics Education Through Simplification
Learning how to simplify logarithmic expressions is a fantastic entry point for deeper mathematical understanding. Grasping these concepts eases the way into more complex topics, enabling students to solve problems efficiently. Engaging with properties like the power rule and product rule provides:
  • A foundation for understanding logarithmic functions.
  • The ability to manipulate and simplify complex equations.
By teaching these fundamental properties, educators equip students with the skills necessary for advanced mathematics. Moreover, simplifying expressions encourages logical thinking and problem-solving, skills that are vital across all areas of education and professional life.

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