5) $$ \log _{4}\left(a^{2} b^{2}\right) $$

Question

UpStudy AI · Accepted Answer

To simplify the logarithmic expression $$ \log _{4}\left(a^{2} b^{2}\right) $$, you can use the properties of logarithms, specifically the **Product Property** and the **Power Property**.

1. **Product Property**: 
   $$
   \log_b (MN) = \log_b M + \log_b N
   $$
   This allows you to separate the product inside the logarithm into the sum of two logarithms.

2. **Power Property**:
   $$
   \log_b (M^k) = k \log_b M
   $$
   This allows you to bring the exponent in front of the logarithm.

Applying these properties step by step:

**Step 1: Apply the Product Property**
$$
\log _{4}\left(a^{2} b^{2}\right) = \log_{4}(a^2) + \log_{4}(b^2)
$$

**Step 2: Apply the Power Property to Each Term**
$$
\log_{4}(a^2) = 2 \log_{4}a
$$
$$
\log_{4}(b^2) = 2 \log_{4}b
$$

**Step 3: Combine the Results**
$$
\log _{4}\left(a^{2} b^{2}\right) = 2 \log_{4}a + 2 \log_{4}b
$$

**Final Simplified Form:**
$$
2 \log_{4}a + 2 \log_{4}b
$$

This is the simplified form of the original logarithmic expression using logarithm properties.
$$ 2 \log_{4}a + 2 \log_{4}b $$

5) \( \log _{4}\left(a^{2} b^{2}\right) \)

Real Tutor Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Algebra Questions