Скачать или смотреть Algebra 2 Practice - Expand a Logarithm (Example 3)

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Expanding a logarithm in Algebra 2 involves applying the *logarithmic properties* to rewrite a single logarithmic expression into multiple terms. This skill is typically practiced to help students simplify complex logarithmic expressions or prepare them for solving equations involving logs.

Key Properties of Logarithms for Expansion:
1. **Product Property**:
\(\log_b(MN) = \log_b(M) + \log_b(N)\)
The log of a product is the sum of the logs.

2. **Quotient Property**:
\(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)
The log of a quotient is the difference of the logs.

3. **Power Property**:
\(\log_b(M^p) = p \cdot \log_b(M)\)
The log of a power brings the exponent to the front.

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Steps to Expand a Logarithmic Expression:
1. *Apply the Product Property* for terms multiplied together.
2. *Apply the Quotient Property* for terms divided.
3. *Apply the Power Property* to simplify exponents.

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Example Problem 1:
Expand \(\log_2(5x^3)\).

#### Solution:
1. Use the Product Property:
\(\log_2(5x^3) = \log_2(5) + \log_2(x^3)\).

2. Apply the Power Property to the second term:
\(\log_2(5) + 3\log_2(x)\).

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Example Problem 2:
Expand \(\log_3\left(\frac{4x^2}{y}\right)\).

#### Solution:
1. Use the Quotient Property:
\(\log_3\left(\frac{4x^2}{y}\right) = \log_3(4x^2) - \log_3(y)\).

2. Apply the Product Property to \(\log_3(4x^2)\):
\(\log_3(4x^2) = \log_3(4) + \log_3(x^2)\).

3. Apply the Power Property to \(\log_3(x^2)\):
\(\log_3(4) + 2\log_3(x) - \log_3(y)\).

Final Answer:
\(\log_3(4) + 2\log_3(x) - \log_3(y)\).

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Tips for Practice:
Be precise with the signs when applying the Quotient Property.
Always check if there are powers or roots to simplify using the Power Property.
For more challenging problems, combine properties in multiple steps.

Would you like a set of practice problems on this topic?

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

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

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