What Is The Value Of This Expression? Log ⁡ 2 8 + Log ⁡ 3 ( 1 3 ) = \log_2 8 + \log_3\left(\frac{1}{3}\right) = Lo G 2 ​ 8 + Lo G 3 ​ ( 3 1 ​ ) = □ \square □

Introduction

In this article, we will explore the value of the expression $\log_2 8 + \log_3\left(\frac{1}{3}\right)$. This expression involves logarithms with different bases, and we will use various properties of logarithms to simplify and evaluate it.

Understanding Logarithms

Before we dive into the expression, let's briefly review the concept of 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 given base is the exponent to which the base must be raised to produce that number.

Properties of Logarithms

There are several important properties of logarithms that we will use to simplify the expression:

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

Simplifying the Expression

Now that we have reviewed the properties of logarithms, let's simplify the expression $\log_2 8 + \log_3\left(\frac{1}{3}\right)$.

First, we can rewrite $\log_2 8$ as $\log_2 (2^3)$ using the power property. This gives us:

$$\log_2 8 = \log_2 (2^3) = 3 \log_2 2$$

Since $\log_a a = 1$ for any base $a$, we know that $\log_2 2 = 1$. Therefore, we have:

$$\log_2 8 = 3 \log_2 2 = 3 \cdot 1 = 3$$

Next, we can rewrite $\log_3\left(\frac{1}{3}\right)$ as $\log_3 (3^{-1})$ using the power property. This gives us:

$$\log_3\left(\frac{1}{3}\right) = \log_3 (3^{-1}) = -1 \log_3 3$$

Since $\log_a a = 1$ for any base $a$, we know that $\log_3 3 = 1$. Therefore, we have:

$$\log_3\left(\frac{1}{3}\right) = -1 \log_3 3 = -1 \cdot 1 = -1$$

Evaluating the Expression

Now that we have simplified the expression, we can evaluate it by adding the two logarithms:

$$\log_2 8 + \log_3\left(\frac{1}{3}\right) = 3 + (-1) = 2$$

Therefore, the value of the expression $\log_2 8 + \log_3\left(\frac{1}{3}\right)$ is $\boxed{2}$.

Conclusion

In this article, we explored the value of the expression $\log_2 8 + \log_3\left(\frac{1}{3}\right)$. We used various properties of logarithms to simplify and evaluate the expression, and we found that its value is $\boxed{2}$. This example illustrates the importance of understanding the properties of logarithms and how to apply them to simplify and evaluate expressions involving logarithms.

Additional Examples

Here are a few additional examples of expressions involving logarithms that you can try to simplify and evaluate on your own:

• $\log_5 25 + \log_2 (1/2)$
• $\log_3 (9) + \log_4 (1/4)$
• $\log_2 (1/8) + \log_3 (27)$

Remember to use the properties of logarithms to simplify and evaluate these expressions, and don't hesitate to ask for help if you get stuck.

References

• [1] "Logarithms" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithms.html
• [2] "Properties of Logarithms" by Purplemath. Retrieved from https://www.purplemath.com/modules/logprops.htm

Glossary

• Logarithm: The inverse operation of exponentiation. If $x = a^y$, then $y = \log_a x$.
• Base: The number to which a logarithm is raised. For example, in the expression $\log_2 8$, the base is 2.
• Exponent: The power to which a number is raised. For example, in the expression $2^3$, the exponent is 3.
• Product Property: $\log_a (xy) = \log_a x + \log_a y$
• Quotient Property: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
• Power Property: $\log_a (x^y) = y \log_a x$
  Q&A: Logarithms and Their Properties =====================================

Introduction

In our previous article, we explored the value of the expression $\log_2 8 + \log_3\left(\frac{1}{3}\right)$. We used various properties of logarithms to simplify and evaluate the expression, and we found that its value is $\boxed{2}$. In this article, we will answer some frequently asked questions about logarithms and their properties.

Q: What is a logarithm?

A: 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 given base is the exponent to which the base must be raised to produce that number.

Q: What are the properties of logarithms?

A: There are several important properties of logarithms that we will use to simplify and evaluate expressions involving logarithms:

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

Q: How do I simplify an expression involving logarithms?

A: To simplify an expression involving logarithms, you can use the properties of logarithms to combine the logarithms into a single logarithm. For example, if you have the expression $\log_2 8 + \log_3\left(\frac{1}{3}\right)$, you can use the product property to combine the logarithms into a single logarithm.

Q: How do I evaluate an expression involving logarithms?

A: To evaluate an expression involving logarithms, you can use the properties of logarithms to simplify the expression and then evaluate the resulting logarithm. For example, if you have the expression $\log_2 8 + \log_3\left(\frac{1}{3}\right)$, you can use the product property to combine the logarithms into a single logarithm and then evaluate the resulting logarithm.

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

A: A logarithm and an exponent are related but distinct concepts. A logarithm is the inverse operation of exponentiation, while an exponent is the power to which a number is raised. For example, if $x = a^y$, then $y = \log_a x$. In this example, $y$ is the exponent, while $\log_a x$ is the logarithm.

Q: Can I use logarithms with any base?

A: Yes, you can use logarithms with any base. However, some bases are more common than others. For example, the base 10 is commonly used in science and engineering, while the base 2 is commonly used in computer science.

Q: How do I convert between different bases?

A: To convert between different bases, you can use the change of base formula. The change of base formula is:

$$\log_b x = \frac{\log_a x}{\log_a b}$$

where $a$ is the new base and $b$ is the original base.

Q: What are some common logarithmic identities?

A: There are several common logarithmic identities that you can use to simplify and evaluate expressions involving logarithms. Some of these identities include:

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

Conclusion

In this article, we answered some frequently asked questions about logarithms and their properties. We covered topics such as the definition of a logarithm, the properties of logarithms, and how to simplify and evaluate expressions involving logarithms. We also discussed some common logarithmic identities and how to convert between different bases.

Additional Resources

• [1] "Logarithms" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithms.html
• [2] "Properties of Logarithms" by Purplemath. Retrieved from https://www.purplemath.com/modules/logprops.htm
• [3] "Change of Base Formula" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f2f7c/logarithms/v/change-of-base-formula

Glossary

• Logarithm: The inverse operation of exponentiation. If $x = a^y$, then $y = \log_a x$.
• Base: The number to which a logarithm is raised. For example, in the expression $\log_2 8$, the base is 2.
• Exponent: The power to which a number is raised. For example, in the expression $2^3$, the exponent is 3.
• Product Property: $\log_a (xy) = \log_a x + \log_a y$
• Quotient Property: $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$
• Power Property: $\log_a (x^y) = y \log_a x$
• Change of Base Formula: $\log_b x = \frac{\log_a x}{\log_a b}$