Use The Properties Of Logarithms To Evaluate Each Of The Following Expressions.(a) 2 Log ⁡ 12 2 + Log ⁡ 12 3 = □ 2 \log_{12} 2 + \log_{12} 3 = \square 2 Lo G 12 ​ 2 + Lo G 12 ​ 3 = □ (b) Ln ⁡ Ε 2 − Ln ⁡ Ε 11 = □ \ln \varepsilon^2 - \ln \varepsilon^{11} = \square Ln Ε 2 − Ln Ε 11 = □

Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will explore how to evaluate logarithmic expressions using the properties of logarithms. We will focus on two specific expressions and demonstrate how to simplify them using the properties of logarithms.

Properties of Logarithms

Before we dive into the examples, let's review the properties of logarithms that we will use to evaluate the expressions.

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$
• Change of Base Property: $\log_b x = \frac{\log_a x}{\log_a b}$

Example (a)

Let's evaluate the expression $2 \log_{12} 2 + \log_{12} 3$.

Step 1: Apply the Power Property

We can rewrite the expression as $\log_{12} 2^2 + \log_{12} 3$ using the power property.

Step 2: Apply the Product Property

We can combine the two logarithmic terms using the product property: $\log_{12} (2^2 \cdot 3)$.

Step 3: Simplify the Expression

We can simplify the expression by evaluating the exponent: $\log_{12} 12$.

Step 4: Evaluate the Logarithm

Since the base and the argument of the logarithm are the same, the value of the logarithm is 1.

Therefore, the final answer is $\boxed{1}$.

Example (b)

Let's evaluate the expression $\ln e^2 - \ln e^{11}$.

Step 1: Apply the Power Property

We can rewrite the expression as $2 \ln e - 11 \ln e$ using the power property.

Step 2: Simplify the Expression

We can simplify the expression by combining the logarithmic terms: $-9 \ln e$.

Step 3: Evaluate the Logarithm

Since the base of the logarithm is $e$, the value of the logarithm is the argument of the logarithm.

Therefore, the final answer is $\boxed{-9}$.

Conclusion

In this article, we demonstrated how to evaluate logarithmic expressions using the properties of logarithms. We focused on two specific expressions and showed how to simplify them using the product, quotient, power, and change of base properties. By understanding and applying these properties, we can simplify complex logarithmic expressions and arrive at the final answer.

Key Takeaways

• The product property states that $\log_b (xy) = \log_b x + \log_b y$.
• The quotient property states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$.
• The power property states that $\log_b x^y = y \log_b x$.
• The change of base property states that $\log_b x = \frac{\log_a x}{\log_a b}$.
• To evaluate logarithmic expressions, we can apply the properties of logarithms to simplify the expression.

Final Thoughts

Introduction

In our previous article, we explored how to evaluate logarithmic expressions using the properties of logarithms. In this article, we will provide a Q&A guide to help readers understand and apply the concepts of logarithmic expressions.

Q: What is a logarithmic expression?

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. Logarithmic expressions are used to solve equations and inequalities that involve exponential functions.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$
• Power Property: $\log_b x^y = y \log_b x$
• Change of Base Property: $\log_b x = \frac{\log_a x}{\log_a b}$

Q: How do I apply the product property?

A: To apply the product property, you need to multiply the arguments of the logarithms. For example, $\log_2 (3 \cdot 4) = \log_2 3 + \log_2 4$.

Q: How do I apply the quotient property?

A: To apply the quotient property, you need to divide the arguments of the logarithms. For example, $\log_2 \left(\frac{3}{4}\right) = \log_2 3 - \log_2 4$.

Q: How do I apply the power property?

A: To apply the power property, you need to multiply the exponent by the logarithm of the base. For example, $\log_2 3^4 = 4 \log_2 3$.

Q: How do I apply the change of base property?

A: To apply the change of base property, you need to divide the logarithm of the argument by the logarithm of the base. For example, $\log_2 3 = \frac{\log_3 3}{\log_3 2}$.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is an expression that involves a logarithm, while an exponential expression is an expression that involves an exponent. For example, $\log_2 3$ is a logarithmic expression, while $2^3$ is an exponential expression.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to apply the properties of logarithms to simplify the expression. For example, $\log_2 (3 \cdot 4) = \log_2 3 + \log_2 4$.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log_2 3$
• $\log_3 4$
• $\log_4 5$
• $\log_5 6$

Conclusion

In this article, we provided a Q&A guide to help readers understand and apply the concepts of logarithmic expressions. We covered the properties of logarithms, how to apply them, and some common logarithmic expressions. By understanding and applying these concepts, readers can simplify complex logarithmic expressions and arrive at the final answer.

Key Takeaways

• The product property states that $\log_b (xy) = \log_b x + \log_b y$.
• The quotient property states that $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$.
• The power property states that $\log_b x^y = y \log_b x$.
• The change of base property states that $\log_b x = \frac{\log_a x}{\log_a b}$.
• To evaluate logarithmic expressions, you need to apply the properties of logarithms to simplify the expression.

Final Thoughts

Logarithmic expressions are a powerful tool in mathematics, and understanding their properties is essential for solving various mathematical problems. By applying the properties of logarithms, we can simplify complex expressions and arrive at the final answer. In this article, we provided a Q&A guide to help readers understand and apply the concepts of logarithmic expressions. We hope that this article has provided valuable insights and knowledge for readers.