Show that the Quotient Rule of Logarithms works for the base 4 expression log4 64−log4 16

Answer 1

To show that the Quotient Rule of Logarithms works for the expression \( \log_4 64 - \log_4 16 \), we need to verify that:

\[ \log_4 64 - \log_4 16 = \log_4 \left( \frac{64}{16} \right) \]

Let's calculate each side step by step.

1. Calculate \( \log_4 64 \):

   • We can rewrite 64 as \( 4^3 \) because \( 4^3 = 64 \).
   • Therefore, \( \log_4 64 = \log_4 (4^3) = 3 \).

2. Calculate \( \log_4 16 \):

   • We can rewrite 16 as \( 4^2 \) because \( 4^2 = 16 \).
   • Therefore, \( \log_4 16 = \log_4 (4^2) = 2 \).

3. Now calculate \( \log_4 64 - \log_4 16 \): \[ \log_4 64 - \log_4 16 = 3 - 2 = 1 \]

4. Calculate \( \frac{64}{16} \): \[ \frac{64}{16} = 4 \]

5. Compute \( \log_4 \left( \frac{64}{16} \right) = \log_4 4 \):

   • Since \( 4 = 4^1 \),
   • Therefore, \( \log_4 4 = 1 \).

Now, both sides are equal:

\[ \log_4 64 - \log_4 16 = 1 \] \[ \log_4 \left( \frac{64}{16} \right) = 1 \]

Thus, we have shown that:

\[ \log_4 64 - \log_4 16 = \log_4 \left( \frac{64}{16} \right) \]

This confirms that the Quotient Rule of Logarithms works for the given expression.