Simplify The Expression:$ \frac{1}{3} \cdot \log _4 8 + \log _4 15 $

Simplify the Expression: A Comprehensive Guide to Logarithmic Equations

In mathematics, logarithmic equations are a fundamental concept that deals with the exponentiation and inverse operations of numbers. These equations are crucial in various fields, including physics, engineering, and computer science. In this article, we will focus on simplifying the expression $\frac{1}{3} \cdot \log _4 8 + \log _4 15$. We will break down the problem step by step, using logarithmic properties and identities to simplify the expression.

Before we dive into the problem, it's essential to understand the basic properties of logarithms. The logarithm of a number $x$ with base $b$ is denoted as $\log_b x$. The logarithmic properties are as follows:

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

Now that we have a good understanding of logarithmic properties, let's simplify the expression $\frac{1}{3} \cdot \log _4 8 + \log _4 15$.

Step 1: Simplify the First Term

The first term is $\frac{1}{3} \cdot \log _4 8$. We can simplify this term using the power rule of logarithms. Since $8 = 2^3$, we can rewrite the term as:

$$\frac{1}{3} \cdot \log _4 8 = \frac{1}{3} \cdot \log _4 (2^3)$$

Using the power rule, we can rewrite the term as:

$$\frac{1}{3} \cdot \log _4 (2^3) = \frac{1}{3} \cdot 3 \cdot \log _4 2$$

Simplifying further, we get:

$$\frac{1}{3} \cdot 3 \cdot \log _4 2 = \log _4 2$$

Step 2: Simplify the Second Term

The second term is $\log _4 15$. We can simplify this term using the change of base formula. Since we want to simplify the expression in terms of base 10, we can use the change of base formula to rewrite the term as:

$$\log _4 15 = \frac{\log_{10} 15}{\log_{10} 4}$$

Step 3: Combine the Terms

Now that we have simplified both terms, we can combine them to get the final expression:

$$\log _4 2 + \frac{\log_{10} 15}{\log_{10} 4}$$

Step 4: Simplify the Expression Further

We can simplify the expression further by combining the two terms. Since the two terms have different bases, we can use the change of base formula to rewrite the expression in terms of a common base. Let's choose base 10 as the common base.

Using the change of base formula, we can rewrite the expression as:

$$\log _4 2 + \frac{\log_{10} 15}{\log_{10} 4} = \log_{10} 2 + \log_{10} 15$$

Using the product rule of logarithms, we can rewrite the expression as:

$$\log_{10} 2 + \log_{10} 15 = \log_{10} (2 \cdot 15)$$

Simplifying further, we get:

$$\log_{10} (2 \cdot 15) = \log_{10} 30$$

In this article, we simplified the expression $\frac{1}{3} \cdot \log _4 8 + \log _4 15$ using logarithmic properties and identities. We broke down the problem step by step, using the product rule, quotient rule, power rule, and change of base formula to simplify the expression. The final simplified expression is $\log_{10} 30$. We hope this article has provided a comprehensive guide to simplifying logarithmic equations.
Frequently Asked Questions: Simplifying Logarithmic Equations

In our previous article, we simplified the expression $\frac{1}{3} \cdot \log _4 8 + \log _4 15$ using logarithmic properties and identities. However, we understand that some readers may still have questions or doubts about the process. In this article, we will address some of the most frequently asked questions about simplifying logarithmic equations.

Q: What is the difference between a logarithm and an exponent?

A: A logarithm is the inverse operation of an exponent. While an exponent raises a number to a power, a logarithm finds the power to which a base must be raised to obtain a given number.

Q: What are the basic properties of logarithms?

A: The basic properties of logarithms are:

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

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the following steps:

1. Identify the base and the argument of the logarithm.
2. Use the product rule, quotient rule, or power rule to simplify the expression.
3. Use the change of base formula to rewrite the expression in terms of a common base.
4. Simplify the expression further by combining like terms.

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

A: A logarithmic equation is an equation that involves a logarithmic expression. For example, $\log_2 x = 3$ is a logarithmic equation, while $\log_2 (x+1)$ is a logarithmic expression.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the following steps:

1. Isolate the logarithmic expression on one side of the equation.
2. Use the change of base formula to rewrite the equation in terms of a common base.
3. Simplify the equation further by combining like terms.
4. Solve for the variable.

Q: What are some common mistakes to avoid when simplifying logarithmic expressions?

A: Some common mistakes to avoid when simplifying logarithmic expressions include:

• Forgetting to use the product rule, quotient rule, or power rule.
• Not using the change of base formula when necessary.
• Simplifying the expression incorrectly.
• Not checking the domain of the logarithmic function.

In this article, we addressed some of the most frequently asked questions about simplifying logarithmic equations. We hope this article has provided a comprehensive guide to simplifying logarithmic expressions and has helped to clarify any doubts or questions you may have had. If you have any further questions or concerns, please don't hesitate to ask.

If you're looking for additional resources to help you learn more about logarithmic equations, here are a few suggestions:

• Khan Academy: Logarithms
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations

We hope this article has been helpful in your understanding of logarithmic equations. Happy learning!