# Expand Logarithms Using The Product Property Worksheet

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Expanding logarithms using the product property means turning one log of a multiplication into several logs added together. For example, $$\log_b(xy)$$ becomes $$\log_b(x) + \log_b(y)$$. This makes tricky log problems easier to handle and solve because we break them into simpler parts. It's like taking a big math problem and splitting it into smaller, more manageable pieces.
Example: Expand $$\log_2(8x)$$ using the product property of logarithms.

Algebra 2
Logarithms

## How Will This Worksheet on "Expand Logarithms Using the Product Property" Benefit Your Student's Learning?

• Splitting logarithms using the product property makes tough math problems easier to handle.
• Helps solve log equations by changing multiplications into easier additions.
• Learning this technique aids in understanding advanced math, such as calculus.
• Reduces errors and simplifies complex problems.
• Enhances logical thinking and problem-solving skills by manipulating mathematical expressions.

## How to Expand Logarithms Using the Product Property?

• Start with a logarithm of a product, such as $$\log_b(xy)$$.
• Use the product property of logarithms, which states $$\log_b(xy) = \log_b(x) + \log_b(y)$$.
• Separate the logarithm into the sum of logarithms of each factor involved in the product.
• Expand the logarithmic expression by writing it as a sum of simpler logarithmic terms, facilitating easier calculation and manipulation.

## Solved Example

Q. Expand the logarithm. Assume all expressions exist and are well-defined. Write your answer as a sum or difference of common logarithms or multiples of common logarithms. The inside of each logarithm must be a distinct constant or variable. $\log uv$
Solution:
1. Identify Property: Identify the property of logarithm used to expand $\log uv$. Product property is used to expand a product within a logarithm.
2. Apply Property: Apply the product property to expand $\log uv$. Product Property: $\log_b (PQ) = \log_b P + \log_b Q$$\newline$ $\log uv = \log(u) + \log(v)$

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