Laws of Logarithms.Let b > 0 b ≠ 1 and let ne R. For u > 0 v > 0 then
1. log_b(uv) = log_b(u) + log_b(v)     (Example: log_2(3x) = log_2(3) + log_2(x) )
2. log_b(u / v) = log_b(u) - ln(v)        (Example: log_3(4/5) = log_3(4) - log_3(5) )

3. log_0(u ^ n) = n * log_0(u)           (Example: log_z(36) = log_z(6 ^ 2) = 2 * log_z(6) )

The **Laws of Logarithms** simplify the manipulation of logarithmic expressions and help in solving logarithmic equations. Let me break down each law correctly with the right notation and examples.

### Corrected Laws of Logarithms:

1. **Product Rule**:  
   \[
   \log_b(uv) = \log_b(u) + \log_b(v)
   \]
   - This states that the logarithm of a product is the sum of the logarithms.
   - **Example**:  
   \[
   \log_2(3x) = \log_2(3) + \log_2(x)
   \]

2. **Quotient Rule**:  
   \[
   \log_b\left(\frac{u}{v}\right) = \log_b(u) - \log_b(v)
   \]
   - This states that the logarithm of a quotient is the difference of the logarithms.
   - **Example**:  
   \[
   \log_3\left(\frac{4}{5}\right) = \log_3(4) - \log_3(5)
   \]

3. **Power Rule**:  
   \[
   \log_b(u^n) = n \cdot \log_b(u)
   \]
   - This states that the logarithm of a power is the exponent multiplied by the logarithm of the base.
   - **Example**:  
   \[
   \log_z(36) = \log_z(6^2) = 2 \cdot \log_z(6)
   \]

### Key Points:
- \( b > 0 \) and \( b \neq 1 \) for the logarithmic base.
- \( u > 0 \), \( v > 0 \) (since the logarithm is undefined for non-positive values).
- \( n \in \mathbb{R} \) (any real number).

These rules are essential for simplifying logarithmic expressions and solving logarithmic equations efficiently.

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Would you like more details on how these rules apply in solving logarithmic equations, or do you have any questions about specific logarithmic properties?

### Related Questions:
1. How can the logarithmic laws help in solving exponential equations?
2. What is the change of base formula for logarithms, and how is it used?
3. How do the logarithmic laws apply when dealing with natural logarithms (\( \ln \))?
4. Can the product and quotient rules be combined in a single expression?
5. How do we solve equations involving both logarithmic and exponential functions?

### Tip:
Always remember that logarithms are only defined for positive numbers, so when simplifying, ensure that the arguments of logarithms are positive.

Laws of Logarithms. Let b > 0, b ≠ 1 and let n ∈ R. For u > 0, v > 0 then 1. log_b(uv) = log_b(u) + log_b(v) (Example: log_2(3x) = log_2(3) + log_2(x)) 2. log_b(u / v) = log_b(u) - log_b(v) (Example: log_3(4/5) = log_3(4) - log_3(5)) 3. log_b(u ^ n) = n * log_b(u) (Example: log_z(36) = log_z(6 ^ 2) = 2 * log_z(6))

Understand the laws of logarithms with clear examples. This guide explains the product, quotient, and power rules of logarithms, including real-life applications. Suitable for students in Grades 10-12.

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