Скачать или смотреть AP Precalculus Practice Test: Unit 2 Question #31 Expand a Single Log into Multiple Logarithms

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To expand a single logarithmic expression into multiple logarithms, you can use the reverse of the logarithmic properties we discussed earlier. Here’s how you can expand a logarithmic expression step-by-step:

Step 1: Use the Product Rule
The *product rule* for logarithms is:

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

This rule allows you to separate the logarithm of a product into the sum of two logarithms.

Step 2: Use the Quotient Rule
The *quotient rule* for logarithms is:

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

This rule allows you to separate the logarithm of a quotient into the difference of two logarithms.

Step 3: Use the Power Rule
The *power rule* for logarithms is:

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

This rule allows you to bring the exponent down as a multiplier in front of the logarithm.

Example 1: Expand \( \log_3(2x) \)

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

So, \( \log_3(2x) \) expands to \( \log_3(2) + \log_3(x) \).

Example 2: Expand \( \log_5\left(\frac{y}{z}\right) \)

1. **Apply the quotient rule**:
\[
\log_5\left(\frac{y}{z}\right) = \log_5(y) - \log_5(z)
\]

So, \( \log_5\left(\frac{y}{z}\right) \) expands to \( \log_5(y) - \log_5(z) \).

Example 3: Expand \( \log_2(x^3y^2) \)

1. **Apply the product rule**:
\[
\log_2(x^3y^2) = \log_2(x^3) + \log_2(y^2)
\]

2. **Apply the power rule to each term**:
\[
\log_2(x^3) = 3 \log_2(x), \quad \log_2(y^2) = 2 \log_2(y)
\]

So, \( \log_2(x^3y^2) \) expands to \( 3 \log_2(x) + 2 \log_2(y) \).

Example 4: Expand \( \log_7\left(\frac{x^4}{y^2}\right) \)

1. **Apply the quotient rule**:
\[
\log_7\left(\frac{x^4}{y^2}\right) = \log_7(x^4) - \log_7(y^2)
\]

2. **Apply the power rule to each term**:
\[
\log_7(x^4) = 4 \log_7(x), \quad \log_7(y^2) = 2 \log_7(y)
\]

So, \( \log_7\left(\frac{x^4}{y^2}\right) \) expands to \( 4 \log_7(x) - 2 \log_7(y) \).

General Steps for Expanding Logs into Multiple Logarithms:
*For a product inside the log**, use the **product rule* to separate it into the sum of two logs.
*For a quotient inside the log**, use the **quotient rule* to separate it into the difference of two logs.
*For an exponent on the argument**, use the **power rule* to bring the exponent in front of the log.

By following these rules, you can expand a single logarithmic expression into multiple logarithms.

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

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