# Condense Logarithms Using The Product Property Worksheet

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Condensing logarithms using the product property involves combining addition logarithmic terms into a single logarithm of a product. For instance, $$\log_b(x) + \log_b(y)$$ condenses to $$\log_b(xy)$$ when the bases are the same. This method simplifies complex logarithmic expressions by reducing them into more compact and manageable forms, which is useful for calculations and problem-solving in various mathematical contexts.
Example: Condense the logarithms $$\log_2(3) + \log_2(5)$$ using the product property.

Algebra 2
Logarithms

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

• Combining logarithms using the product property simplifies complex math problems by consolidating terms.
• Adding logarithms into one expression makes calculations quicker and easier.
• Understanding how to condense logarithms helps students handle more challenging math problems effectively.
• Prepares students for advanced math topics like calculus and algebra by building a strong foundation.
• Reduces the chance of errors in calculations, ensuring accuracy.

## How to Condense Logarithms Using the Product Property?

• Begin with separate logarithmic terms, such as $$\log_b(x)$$ and $$\log_b(y)$$.
• Combine the separate logarithms using the product property, which states $$\log_b(x) + \log_b(y) = \log_b(xy)$$, ensuring the bases are the same.
• Multiply the arguments inside the logarithm to condense the expression into a single logarithm of their product.
• Ensure that the condensed logarithm accurately represents the original separate logarithmic terms combined into a single expression.

## Solved Example

Q. Condense the logarithm. Assume all expressions exist and are well-defined.$\newline$$\log_a 5 + \log_a 3$
Solution:
1. Identify Property: Identify the property of logarithm used to condense $\log_a 5 + \log_a 3$.$\newline$Since these logarithms have the same base, we can apply the product rule to condense the logarithms:
2. Apply Sum Property: Apply the product property to condense $\log_a 5 + \log_a 3$.$\newline$ $\log_b P + \log_b Q = \log_b (PQ)$$\newline$$\log_a 5 + \log_a 3 = \log_a (5 \cdot 3)$
3. Simplify Expression: Simplify the expression inside the logarithm.$\newline$$\log_a (5 \cdot 3) = \log_a 15$

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