What Is The Value Of This Expression? Log ⁡ 2 8 + Log ⁡ 3 ( 1 3 ) = □ \log _2 8+\log _3\left(\frac{1}{3}\right) = \square Lo G 2 ​ 8 + Lo G 3 ​ ( 3 1 ​ ) = □ Type The Correct Answer In The Box. Use Numerals Instead Of Words.

Introduction

When dealing with logarithmic expressions, it's essential to understand the properties and rules that govern them. In this article, we will explore the value of the expression $\log _2 8+\log _3\left(\frac{1}{3}\right)$. To find the value of this expression, we need to apply the properties of logarithms, specifically the product rule and the change of base formula.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$, then $y = \log_a x$. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. For example, $\log_2 8 = 3$ because $2^3 = 8$.

The Product Rule

The product rule of logarithms states that $\log_a (xy) = \log_a x + \log_a y$. This rule allows us to combine the logarithms of two numbers with the same base into a single logarithm.

The Change of Base Formula

The change of base formula states that $\log_a x = \frac{\log_b x}{\log_b a}$. This formula allows us to change the base of a logarithm from one base to another.

Applying the Product Rule and Change of Base Formula

To find the value of the expression $\log _2 8+\log _3\left(\frac{1}{3}\right)$, we can apply the product rule and the change of base formula.

First, let's simplify the expression $\log _3\left(\frac{1}{3}\right)$. Using the change of base formula, we can rewrite this expression as $\frac{\log_2 \left(\frac{1}{3}\right)}{\log_2 3}$.

Simplifying the Expression

Now, let's simplify the expression $\log_2 \left(\frac{1}{3}\right)$. We can rewrite this expression as $\log_2 1 - \log_2 3$. Using the property of logarithms that $\log_a 1 = 0$, we can simplify this expression to $-\log_2 3$.

Substituting the Simplified Expression

Now, let's substitute the simplified expression $-\log_2 3$ back into the original expression. We get $\log _2 8 - \log_2 3$.

Applying the Product Rule

Using the product rule, we can rewrite the expression $\log _2 8 - \log_2 3$ as $\log_2 \left(\frac{8}{3}\right)$.

Finding the Value of the Expression

Now, let's find the value of the expression $\log_2 \left(\frac{8}{3}\right)$. We can rewrite this expression as $\log_2 2^3 - \log_2 3$. Using the property of logarithms that $\log_a a^x = x$, we can simplify this expression to $3 - \log_2 3$.

Finding the Value of $\log_2 3$

To find the value of the expression $3 - \log_2 3$, we need to find the value of $\log_2 3$. Unfortunately, this value is not a rational number, and it cannot be expressed as a finite decimal. However, we can approximate this value using a calculator or a computer program.

Approximating the Value of $\log_2 3$

Using a calculator or a computer program, we can approximate the value of $\log_2 3$ to be approximately 1.58496.

Finding the Value of the Expression

Now, let's substitute the approximate value of $\log_2 3$ back into the expression $3 - \log_2 3$. We get $3 - 1.58496$, which is approximately 1.41504.

Conclusion

In conclusion, the value of the expression $\log _2 8+\log _3\left(\frac{1}{3}\right)$ is approximately 1.41504.

Final Answer

The final answer is: 1.41504

Introduction

In the previous article, we explored the value of the expression $\log _2 8+\log _3\left(\frac{1}{3}\right)$. In this article, we will answer some frequently asked questions (FAQs) about logarithmic expressions.

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

A: A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$, then $y = \log_a x$. An exponent is a number that is raised to a power.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_a (xy) = \log_a x + \log_a y$. This rule allows us to combine the logarithms of two numbers with the same base into a single logarithm.

Q: What is the change of base formula?

A: The change of base formula states that $\log_a x = \frac{\log_b x}{\log_b a}$. This formula allows us to change the base of a logarithm from one base to another.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms, such as the product rule and the change of base formula. You can also use the fact that $\log_a 1 = 0$ and $\log_a a = 1$.

Q: What is the value of $\log_2 8$?

A: The value of $\log_2 8$ is 3, because $2^3 = 8$.

Q: What is the value of $\log_3 \left(\frac{1}{3}\right)$?

A: The value of $\log_3 \left(\frac{1}{3}\right)$ is -1, because $3^{-1} = \frac{1}{3}$.

Q: How do I find the value of a logarithmic expression?

A: To find the value of a logarithmic expression, you can use the properties of logarithms, such as the product rule and the change of base formula. You can also use a calculator or a computer program to approximate the value of the expression.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a finite decimal or a fraction. An irrational number is a number that cannot be expressed as a finite decimal or a fraction.

Q: Is the value of $\log_2 3$ a rational number or an irrational number?

A: The value of $\log_2 3$ is an irrational number, because it cannot be expressed as a finite decimal or a fraction.

Q: How do I approximate the value of an irrational number?

A: To approximate the value of an irrational number, you can use a calculator or a computer program.

Q: What is the value of the expression $\log _2 8+\log _3\left(\frac{1}{3}\right)$?

A: The value of the expression $\log _2 8+\log _3\left(\frac{1}{3}\right)$ is approximately 1.41504.

Conclusion

In conclusion, logarithmic expressions can be simplified using the properties of logarithms, such as the product rule and the change of base formula. The value of a logarithmic expression can be found using a calculator or a computer program. We hope that this article has been helpful in answering your questions about logarithmic expressions.

Final Answer

The final answer is: 1.41504