Write The Following Expression In Condensed Form:$ \log(8) - \frac{1}{2} \log(x) }$Options A. { \log \left(\frac{8 {x^2}\right)$}$ B. { \log \left(\frac{8}{\sqrt{x}}\right)$}$ C. { \log \left(8 X^2\right)$}$

Introduction

Logarithmic expressions can be complex and difficult to simplify, but with the right techniques, they can be condensed into more manageable forms. In this article, we will explore how to simplify the expression $\log(8) - \frac{1}{2} \log(x)$ using various logarithmic properties.

Understanding Logarithmic Properties

Before we dive into simplifying the expression, it's essential to understand the properties of logarithms. The two main properties we will use are:

• Product Property: $\log(a \cdot b) = \log(a) + \log(b)$
• Quotient Property: $\log(\frac{a}{b}) = \log(a) - \log(b)$
• Power Property: $\log(a^b) = b \cdot \log(a)$

Simplifying the Expression

Now that we have a good understanding of logarithmic properties, let's simplify the expression $\log(8) - \frac{1}{2} \log(x)$.

Step 1: Rewrite the Expression

We can start by rewriting the expression using the power property. Since $\frac{1}{2}$ is the same as $x^{-\frac{1}{2}}$, we can rewrite the expression as:

$$\log(8) - \log(x^{\frac{1}{2}})$$

Step 2: Apply the Quotient Property

Now that we have rewritten the expression, we can apply the quotient property to simplify it further:

$$\log(8) - \log(x^{\frac{1}{2}}) = \log\left(\frac{8}{x^{\frac{1}{2}}}\right)$$

Step 3: Simplify the Expression

We can simplify the expression further by recognizing that $x^{\frac{1}{2}}$ is the same as $\sqrt{x}$. Therefore, we can rewrite the expression as:

$$\log\left(\frac{8}{x^{\frac{1}{2}}}\right) = \log\left(\frac{8}{\sqrt{x}}\right)$$

Conclusion

In this article, we simplified the expression $\log(8) - \frac{1}{2} \log(x)$ using various logarithmic properties. We started by rewriting the expression using the power property, then applied the quotient property to simplify it further. Finally, we simplified the expression by recognizing that $x^{\frac{1}{2}}$ is the same as $\sqrt{x}$. The simplified expression is $\log\left(\frac{8}{\sqrt{x}}\right)$.

Comparison with Options

Now that we have simplified the expression, let's compare it with the options provided:

• Option A: $\log \left(\frac{8}{x^2}\right)$
• Option B: $\log \left(\frac{8}{\sqrt{x}}\right)$
• Option C: $\log \left(8 x^2\right)$

As we can see, the simplified expression $\log\left(\frac{8}{\sqrt{x}}\right)$ matches Option B.

Final Answer

Introduction

In our previous article, we explored how to simplify the expression $\log(8) - \frac{1}{2} \log(x)$ using various logarithmic properties. In this article, we will answer some frequently asked questions related to simplifying logarithmic expressions.

Q&A

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log(a \cdot b) = \log(a) + \log(b)$. This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do I apply the quotient property of logarithms?

A: To apply the quotient property of logarithms, you need to subtract the logarithm of the divisor from the logarithm of the dividend. For example, $\log(\frac{a}{b}) = \log(a) - \log(b)$.

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log(a^b) = b \cdot \log(a)$. This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, you need to apply the product property and the quotient property in the correct order. For example, $\log(a) + \log(b) - \log(c) = \log(a \cdot b) - \log(c) = \log(\frac{a \cdot b}{c})$.

Q: Can I simplify a logarithmic expression with a negative exponent?

A: Yes, you can simplify a logarithmic expression with a negative exponent by applying the power property. For example, $\log(a^{-b}) = -b \cdot \log(a)$.

Q: How do I evaluate a logarithmic expression with a variable base?

A: To evaluate a logarithmic expression with a variable base, you need to use the change of base formula. For example, $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, where $c$ is any positive real number.

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, $\log(a)$ is a logarithmic expression, while $a^b$ is an exponential expression.

Conclusion

In this article, we answered some frequently asked questions related to simplifying logarithmic expressions. We covered topics such as the product property, quotient property, power property, and change of base formula. We also discussed how to simplify logarithmic expressions with multiple terms, negative exponents, and variable bases.

Final Tips

• Always apply the logarithmic properties in the correct order to avoid errors.
• Use the change of base formula to evaluate logarithmic expressions with variable bases.
• Simplify logarithmic expressions by applying the product property, quotient property, and power property.
• Be careful when working with negative exponents and variable bases.

Common Mistakes

• Failing to apply the logarithmic properties in the correct order.
• Not using the change of base formula when evaluating logarithmic expressions with variable bases.
• Simplifying logarithmic expressions incorrectly by applying the wrong logarithmic property.
• Not being careful when working with negative exponents and variable bases.

Final Answer

The final answer is that simplifying logarithmic expressions requires a good understanding of the logarithmic properties and a careful application of these properties. By following the tips and avoiding common mistakes, you can simplify logarithmic expressions with confidence.