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

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 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: $\log \left(\frac{a}{b}\right) = \log (a) - \log (b)$
• Power Property: $\log (a^b) = b \cdot \log (a)$

Simplifying the Expression

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

Step 1: Apply the Power Property

The first step is to apply the power property to the second term, $\frac{1}{2} \log (x)$. This can be rewritten as $\log (x^{\frac{1}{2}})$.

import math
log_8 = math.log(8)
log_x_half = math.log(math.sqrt(x))

log_x_half = math.log(x ** 0.5)

Step 2: Apply the Quotient Property

Now that we have $\log (8) - \log (x^{\frac{1}{2}})$, we can apply the quotient property to simplify the expression further.

# Apply the quotient property
log_8_minus_log_x_half = math.log(8) - math.log(math.sqrt(x))

Step 3: Simplify the Expression

Using the quotient property, we can rewrite the expression as $\log \left(\frac{8}{x^{\frac{1}{2}}}\right)$.

# Simplify the expression
simplified_expression = math.log(8 / math.sqrt(x))

Step 4: Rationalize the Denominator

To rationalize the denominator, we can multiply the numerator and denominator by $\sqrt{x}$.

# Rationalize the denominator
rationalized_expression = math.log((8 * math.sqrt(x)) / x)

Step 5: Simplify the Expression Further

Now that we have $\log \left(\frac{8 \sqrt{x}}{x}\right)$, we can simplify the expression further by canceling out the common factors.

# Simplify the expression further
final_expression = math.log(8 * math.sqrt(x) / x)

Conclusion

In this article, we simplified the expression $\log (8) - \frac{1}{2} \log (x)$ using logarithmic properties. We applied the power property, quotient property, and rationalized the denominator to arrive at the final expression $\log \left(\frac{8 \sqrt{x}}{x}\right)$. This expression can be further simplified by canceling out the common factors.

Answer

The correct answer is:

• B. $\log \left(\frac{8}{\sqrt{x}}\right)$

Note that the final expression we arrived at is not among the answer choices. However, we can simplify the expression further by canceling out the common factors, which would result in the correct answer.

Discussion

The expression $\log (8) - \frac{1}{2} \log (x)$ can be simplified using logarithmic properties. The power property allows us to rewrite the second term as $\log (x^{\frac{1}{2}})$. The quotient property then enables us to simplify the expression further. Finally, rationalizing the denominator and canceling out common factors result in the final expression $\log \left(\frac{8 \sqrt{x}}{x}\right)$.

Example Use Cases

Logarithmic expressions are commonly used in various fields, including mathematics, physics, and engineering. Here are a few example use cases:

• Sound Level Measurement: Logarithmic expressions are used to measure sound levels in decibels (dB). The sound level is calculated as $L = 10 \log \left(\frac{I}{I_0}\right)$, where $I$ is the intensity of the sound and $I_0$ is a reference intensity.
• Electrical Circuits: Logarithmic expressions are used to analyze electrical circuits. The voltage across a resistor is given by $V = I R \log \left(\frac{R}{R_0}\right)$, where $I$ is the current, $R$ is the resistance, and $R_0$ is a reference resistance.
• Finance: Logarithmic expressions are used in finance to calculate returns on investment. The return on investment is given by $R = \log \left(\frac{P}{P_0}\right)$, where $P$ is the current price and $P_0$ is the initial price.

Conclusion

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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 common questions and answers related to logarithmic expressions.

Q: What is a logarithmic expression?

A: A logarithmic expression is an expression that involves logarithms, which are the inverse of exponentials. Logarithmic expressions are used to represent quantities that are proportional to the logarithm of another quantity.

Q: What are the common logarithmic properties?

A: The common logarithmic properties are:

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

Q: How do I simplify a logarithmic expression?

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

1. Apply the power property: Rewrite the expression using the power property.
2. Apply the quotient property: Rewrite the expression using the quotient property.
3. Rationalize the denominator: Multiply the numerator and denominator by the conjugate of the denominator.
4. Simplify the expression: Cancel out common factors and simplify the expression.

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

A: A logarithmic expression is an expression that involves logarithms, while an exponential expression is an expression that involves exponents. For example, $\log (x)$ is a logarithmic expression, while $x^2$ is an exponential expression.

Q: How do I evaluate a logarithmic expression?

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

1. Check if the expression is in the form $\log (a)$: If the expression is in the form $\log (a)$, you can evaluate it by finding the value of $a$.
2. Check if the expression is in the form $\log (a) + \log (b)$: If the expression is in the form $\log (a) + \log (b)$, you can evaluate it by finding the values of $a$ and $b$ and adding their logarithms.
3. Check if the expression is in the form $\log (a) - \log (b)$: If the expression is in the form $\log (a) - \log (b)$, you can evaluate it by finding the values of $a$ and $b$ and subtracting their logarithms.

Q: What are some common applications of logarithmic expressions?

A: Logarithmic expressions have many common applications in various fields, including:

• Sound level measurement: Logarithmic expressions are used to measure sound levels in decibels (dB).
• Electrical circuits: Logarithmic expressions are used to analyze electrical circuits.
• Finance: Logarithmic expressions are used in finance to calculate returns on investment.

Conclusion

In conclusion, logarithmic expressions can be complex and difficult to simplify, but with the right techniques, they can be condensed into more manageable forms. By understanding the common logarithmic properties and applying the power property, quotient property, and rationalizing the denominator, you can simplify logarithmic expressions and evaluate them. Logarithmic expressions have many common applications in various fields, including sound level measurement, electrical circuits, and finance.