write this using properites of logs log6[(a/b)^4 x c]

To express \( \log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right] \) using the properties of logarithms, follow these steps:

1. **Apply the logarithm to the product**:
   \[
   \log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right] = \log_6\left[\left(\frac{a}{b}\right)^4\right] + \log_6(c)
   \]

2. **Apply the logarithm to the quotient inside**:
   \[
   \log_6\left[\left(\frac{a}{b}\right)^4\right] = 4 \cdot \log_6\left(\frac{a}{b}\right)
   \]

3. **Break down the logarithm of the quotient**:
   \[
   \log_6\left(\frac{a}{b}\right) = \log_6(a) - \log_6(b)
   \]

4. **Combine everything**:
   \[
   \log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right] = 4\left(\log_6(a) - \log_6(b)\right) + \log_6(c)
   \]

So, the final expression is:
\[
\log_6\left[\left(\frac{a}{b}\right)^4 \cdot c\right] = 4\log_6(a) - 4\log_6(b) + \log_6(c)
\]

Would you like any further explanation or have questions on this?

Here are five related questions:
1. How can you express \(\log_b(x \cdot y)\) using properties of logarithms?
2. What is the general rule for the logarithm of a power, like \(\log_b(x^n)\)?
3. How would you simplify \(\log_6\left(a^2 \cdot b^3\right)\)?
4. Can you explain how to apply the change of base formula to logarithms?
5. What happens if the base of the logarithm matches the argument, such as \(\log_6(6)\)?

**Tip:** Remember that the logarithmic properties allow you to break down complex expressions into simpler terms, which is helpful when solving logarithmic equations!

write this using properties of logs log6[(a/b)^4 x c]

Learn how to simplify the expression log6[(a/b)^4 * c] using the properties of logarithms. This problem explores key concepts in algebra, including applying logarithms to products, quotients, and powers. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation