Math Problem Statement

Write the expression 2 log_b 6 - log_b 2 as a single logarithm in simplest form.

Solution

To simplify the expression 2logb6logb22 \log_b 6 - \log_b 2 as a single logarithm, let's follow these steps:

  1. Apply the Power Rule: The term 2logb62 \log_b 6 can be rewritten using the power rule for logarithms, alogbx=logb(xa)a \log_b x = \log_b (x^a). 2logb6=logb(62)=logb362 \log_b 6 = \log_b (6^2) = \log_b 36

    So the expression now becomes: logb36logb2\log_b 36 - \log_b 2

  2. Apply the Quotient Rule: Now, use the quotient rule for logarithms, which states logbxlogby=logb(xy)\log_b x - \log_b y = \log_b \left( \frac{x}{y} \right). logb36logb2=logb(362)\log_b 36 - \log_b 2 = \log_b \left( \frac{36}{2} \right)

  3. Simplify the Quotient: 362=18\frac{36}{2} = 18

    So the expression simplifies to: logb18\log_b 18

Final Answer:

2logb6logb2=logb182 \log_b 6 - \log_b 2 = \log_b 18

Would you like further details on any of these steps?

Here are some related questions to consider:

  1. How do you apply the power rule for logarithms in other contexts?
  2. What is the quotient rule for logarithms, and when is it useful?
  3. Can you combine different logarithmic expressions with other bases?
  4. How do you expand or break down logarithmic expressions with multiple terms?
  5. What happens if you add instead of subtract two logarithmic expressions?

Tip: Always look for opportunities to apply the power, product, or quotient rules to simplify logarithmic expressions effectively.

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Math Problem Analysis

Mathematical Concepts

Logarithmic Expressions
Algebra

Formulas

Power Rule of Logarithms: a * log_b(x) = log_b(x^a)
Quotient Rule of Logarithms: log_b(x) - log_b(y) = log_b(x/y)

Theorems

Logarithmic Properties

Suitable Grade Level

Grades 9-12