# Identify Properties Of Logarithms Given Equation Worksheet

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To identify properties of logarithms given an equation, look for ways to apply logarithmic rules such as the product rule $$\log_b(xy) = \log_b(x) + \log_b(y)$$, the quotient rule \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y), and the power rule $$\log_b(x^k) = k \log_b(x)$$. Recognizing these properties allows us to simplify and solve logarithmic equations more effectively.

Example: Given the equation $$\log_b(x^2) = 2 \log_b(x)$$, identify the logarithmic property used.

Algebra 2
Logarithms

## How Will This Worksheet on "Identify Properties of Logarithms Given Equation" Benefit Your Student's Learning?

• Understanding logarithmic properties allows students to simplify tough equations faster, which is crucial during exams and assignments.
• Mastery of logarithmic relationships helps students grasp concepts more intuitively, making problem-solving easier and more effective.
• Knowing logarithmic properties well sets a strong base for tackling higher-level math subjects like calculus and advanced algebra.
• Applying logarithmic rules enhances analytical thinking and logical reasoning, which are vital for solving complex problems in various subjects.

## How to Identify Properties of Logarithms Given Equation?

• First, identify if the equation follows logarithmic rules such as the product, quotient, or power rules.
• Confirm the base b used in all logarithmic terms within the equation.
• Then, combine the logarithmic terms using the identified properties to reduce the equation to its simplest logarithmic expression.
• Confirm that the simplified expression on the left side matches the right side.
• Simplify the logarithmic expression further if possible, ensuring clarity and correctness in the equation.

## Solved Example

Q. Which property of logarithms does this equation demonstrate? $\newline$$\log_3 3 + \log_3 6 = \log_3 18$$\newline$Choices:$\newline$(A) $\text{Product Property}$$\newline$(B) $\text{Power Property}$$\newline$(C) $\text{Quotient Property}$
Solution:
1. Analyze the equation: Analyze the given equation.$\newline$We have the equation $\log_3 3 + \log_3 6 = \log_3 18$. We need to determine which logarithmic property this equation represents.
2. Recall logarithmic properties: Recall the properties of logarithms.$\newline$There are three main properties of logarithms that are relevant to this problem: the Product Property, the Power Property, and the Quotient Property. The Product Property states that $\log_b (P) + \log_b (Q) = \log_b (PQ)$, the Power Property states that $n \cdot \log_b (P) = \log_b (P^n)$, and the Quotient Property states that $\log_b (P) - \log_b (Q) = \log_b \left(\frac{P}{Q}\right)$.
3. Match equation with property: Match the given equation with the correct property.$\newline$The given equation is $\log_3 3 + \log_3 6 = \log_3 18$. This matches the form of the Product Property, which states that the sum of the logarithms is equal to the logarithm of the product of the bases: $\log_b (P) + \log_b (Q) = \log_b (PQ)$.
4. Verify equation using property: Verify the equation using the Product Property.$\newline$Using the Product Property, we can combine the logarithms on the left side of the equation: $\log_3 (3 \times 6) = \log_3 18$. Since $3 \times 6$ equals $18$, the equation is correct and demonstrates the Product Property.

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