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Chapter 10: Problem 69

Write as a single logarithm. Assume \(x>0 .\) \(\log (x+2)+\log (x+3)\)

Short Answer

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

Step by step solution

01

Use the Product Rule of Logarithms

Identify that the expression \(\log (x+2)+\log (x+3)\) can be simplified using the product rule of logarithms which states that \(\log(a) + \log(b) = \log(ab)\).
02

Apply the Product Rule

Combine the logarithms using the product rule: \(\log(x+2) + \log(x+3) = \log((x+2)(x+3))\).
03

Write the Final Expression

After applying the product rule, the expression simplifies to \(\log((x+2)(x+3))\).

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Key Concepts

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

product rule of logarithms
One of the powerful tools when dealing with logarithmic expressions is the product rule of logarithms. This rule states that the sum of two logarithms can be combined into a single logarithm. Specifically, the product rule is written as follows: \(\log(a) + \log(b) = \log(ab)\).

This rule works because logarithms are the inverses of exponents. When you add two logarithms together, you're effectively multiplying the two original numbers. Imagine you have \(\log(x+2) + \log(x+3)\). By applying the product rule, you simplify it to \(\log((x+2)(x+3))\).

Understanding this rule can significantly streamline solving logarithmic expressions.
logarithm properties
Logarithms have several important properties that can help simplify complex expressions. Here are the main properties:
  • Product Rule: \(\log(a) + \log(b) = \log(ab)\)
  • Quotient Rule: \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\)
  • Power Rule: \(\log(a^b) = b \log(a)\)
  • Change of Base Formula: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\)


These properties allow for the transformation of logarithmic expressions into more manageable forms. For our specific task, understanding the product rule was crucial in combining the two logarithms into one.
combining logarithms
Combining logarithms is a technique used to simplify expressions with multiple logarithms. By leveraging the properties of logarithms鈥攅specially the product, quotient, and power rules鈥攚e can rewrite these expressions more concisely.

In the given exercise, we needed to combine \(\log(x+2)\) and \(\log(x+3)\). Using the product rule, we merged them into a single logarithm: \(\log((x+2)(x+3))\).

When combining logarithms:
  • Identify which property applies (e.g., product, quotient, or power rule).
  • Apply the rule correctly to simplify the expression.
  • Rewrite the expression in its simplest form.


By mastering these techniques, handling logarithmic expressions becomes a much simpler task.

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Most popular questions from this chapter

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