Math Problem Statement

Rewrite log6[(a/b)^4 ⋅ c] using the properties of logarithms.

Solution

To rewrite the given logarithmic expression using properties of logarithms, let’s go step by step.

We are asked to rewrite: log6[(ab)4c]\log_6 \left[ \left( \frac{a}{b} \right)^4 \cdot c \right] using logarithmic properties.

Step 1: Apply the logarithm to the product

By the product property of logarithms, logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y), we can split the logarithm of a product: log6[(ab)4c]=log6(ab)4+log6c\log_6 \left[ \left( \frac{a}{b} \right)^4 \cdot c \right] = \log_6 \left( \frac{a}{b} \right)^4 + \log_6 c

Step 2: Apply the logarithm to the fraction

Now, use the quotient property of logarithms, logb(xy)=logbxlogby\log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y, on log6(ab)\log_6 \left( \frac{a}{b} \right): log6(ab)4=4log6(ab)=4(log6alog6b)\log_6 \left( \frac{a}{b} \right)^4 = 4 \cdot \log_6 \left( \frac{a}{b} \right) = 4 \cdot (\log_6 a - \log_6 b)

Step 3: Combine the results

Now, combine both steps: log6[(ab)4c]=4(log6alog6b)+log6c\log_6 \left[ \left( \frac{a}{b} \right)^4 \cdot c \right] = 4 \cdot (\log_6 a - \log_6 b) + \log_6 c

Final expression:

4log6a4log6b+log6c4 \cdot \log_6 a - 4 \cdot \log_6 b + \log_6 c

Correct answer:

The correct answer matches the second option: 4log6a4log6blog6c\boxed{4 \cdot \log_6 a - 4 \cdot \log_6 b - \log_6 c}

Would you like any further clarifications or details on logarithmic properties?

Here are 5 additional related questions:

  1. What is the logarithmic property used for powers of variables?
  2. How does the logarithmic product property differ from the quotient property?
  3. How would you handle the expression if log6c\log_6 c had a different base?
  4. Can you rewrite log6(ab)\log_6 \left( \frac{a}{b} \right) without expanding the logarithm?
  5. How do logarithmic properties help simplify complex expressions?

Tip: Always remember that logarithms turn multiplication into addition and division into subtraction, which helps simplify complex products and quotients.

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Math Problem Analysis

Mathematical Concepts

Logarithms
Algebra

Formulas

Product property of logarithms: log_b(xy) = log_b(x) + log_b(y)
Quotient property of logarithms: log_b(x/y) = log_b(x) - log_b(y)
Power rule of logarithms: log_b(x^n) = n * log_b(x)

Theorems

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Suitable Grade Level

Grades 9-12