Rewrite log6[(a/b)^4 x c]

Question

using the properties of logarithms.

Answer 1

To rewrite the expression \( \log_6\left(\frac{a}{b}\right)^4 \cdot c \) using the properties of logarithms, we can apply the following properties:

1. \( \log_b(m^n) = n \cdot \log_b(m) \)
2. \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \)

Now, let's rewrite the expression step by step:

1. Start with \( \log_6\left(\left(\frac{a}{b}\right)^4 \cdot c\right) \).

2. Apply the product property: \[ \log_6\left(\left(\frac{a}{b}\right)^4\right) + \log_6(c) \]

3. Now apply the power property to the first term: \[ 4 \cdot \log_6\left(\frac{a}{b}\right) + \log_6(c) \]

4. Finally, use the quotient property on \( \log_6\left(\frac{a}{b}\right) \): \[ 4 \cdot \left( \log_6(a) - \log_6(b) \right) + \log_6(c) \]

Putting it all together, we have:

\[ 4 \log_6(a) - 4 \log_6(b) + \log_6(c) \]

This is the final expression using the properties of logarithms.