AP Precalculus Practice Test: Unit 2 Question #30 Simplify Logs into a Single Logarithm

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To simplify logarithmic expressions into a single logarithm, you can use the logarithmic properties. Here’s an outline of the process for simplifying logs into a single logarithm:

### Step 1: Use the Product Rule
The product rule for logarithms states that:

\[
\log_b(x) + \log_b(y) = \log_b(xy)
\]

This rule is used when you have the sum of two logarithms with the same base. You can combine them into the logarithm of the product of their arguments.

### Step 2: Use the Quotient Rule
The quotient rule for logarithms states that:

\[
\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)
\]

This rule is used when you have the difference between two logarithms with the same base. You can combine them into the logarithm of the quotient of their arguments.

### Step 3: Use the Power Rule
The power rule for logarithms states that:

\[
a \log_b(x) = \log_b(x^a)
\]

This rule is used when you have a logarithmic expression where the argument has an exponent. You can move the exponent to the front as a multiplier and simplify the logarithmic expression.

### Example 1: Simplify \( \log_3(5) + \log_3(2) \)

1. **Apply the product rule**:
\[
\log_3(5) + \log_3(2) = \log_3(5 \times 2) = \log_3(10)
\]

So, \( \log_3(5) + \log_3(2) \) simplifies to \( \log_3(10) \).

### Example 2: Simplify \( \log_4(8) - \log_4(2) \)

1. **Apply the quotient rule**:
\[
\log_4(8) - \log_4(2) = \log_4\left(\frac{8}{2}\right) = \log_4(4)
\]

2. **Simplify the result**:
\[
\log_4(4) = 1 \quad \text{(since \( 4^1 = 4 \))}
\]

So, \( \log_4(8) - \log_4(2) \) simplifies to \( 1 \).

### Example 3: Simplify \( 3 \log_2(4) \)

1. **Apply the power rule**:
\[
3 \log_2(4) = \log_2(4^3) = \log_2(64)
\]

So, \( 3 \log_2(4) \) simplifies to \( \log_2(64) \).

### General Steps for Simplifying Logs into a Single Logarithm:
- **If the expression is a sum of logs**, apply the **product rule**.
- **If the expression is a difference of logs**, apply the **quotient rule**.
- **If the expression has a logarithm multiplied by a constant**, apply the **power rule**.

By following these steps and using the appropriate rules, you can simplify logarithmic expressions into a single logarithm.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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