Rewrite The Logarithmic Expression As A Single Logarithm With The Same Base. Assume All Expressions Exist And Are Well-defined. Simplify Any Fractions.${\log_z 12 - 7\log_z 2}$

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and rewriting them as a single logarithm with the same base is an essential skill for any math enthusiast. In this article, we will explore how to rewrite the given logarithmic expression as a single logarithm with the same base, and simplify any fractions that may arise.

Understanding Logarithmic Expressions

Before we dive into rewriting the logarithmic expression, let's first understand what logarithmic expressions are. A logarithmic expression is an expression that involves the logarithm of a number. The logarithm of a number is the power to which a base number must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because 10^2 = 100.

Rewriting the Logarithmic Expression

The given logarithmic expression is $\log_z 12 - 7\log_z 2$. To rewrite this expression as a single logarithm with the same base, we can use the properties of logarithms. Specifically, we can use the property that states $\log_a b - \log_a c = \log_a \frac{b}{c}$.

Using this property, we can rewrite the given expression as follows:

$\log_z 12 - 7\log_z 2 = \log_z \frac{12}{2^7}$

Simplifying the Fraction

Now that we have rewritten the logarithmic expression as a single logarithm with the same base, we can simplify the fraction that arises. The fraction $\frac{12}{2^7}$ can be simplified as follows:

$\frac{12}{2^7} = \frac{12}{128} = \frac{3}{32}$

Therefore, the rewritten logarithmic expression is:

$\log_z \frac{3}{32}$

Conclusion

In this article, we have learned how to rewrite the given logarithmic expression as a single logarithm with the same base, and simplify any fractions that may arise. We have used the properties of logarithms to rewrite the expression, and simplified the fraction that arises. This is an essential skill for any math enthusiast, and will be useful in a variety of mathematical applications.

Properties of Logarithms

Before we conclude, let's review some of the properties of logarithms that we have used in this article. These properties are essential for working with logarithmic expressions, and are summarized below:

• $\log_a b - \log_a c = \log_a \frac{b}{c}$
• $\log_a b + \log_a c = \log_a (bc)$
• $\log_a b^c = c\log_a b$

These properties can be used to rewrite logarithmic expressions in a variety of ways, and are essential for working with logarithmic functions.

Real-World Applications

Logarithmic expressions have a wide range of real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic expressions are used to calculate the power of a signal and the gain of an amplifier.

In conclusion, rewriting logarithmic expressions as a single logarithm with the same base is an essential skill for any math enthusiast. By using the properties of logarithms, we can rewrite logarithmic expressions in a variety of ways, and simplify any fractions that may arise. This is a fundamental concept in mathematics, and will be useful in a variety of mathematical applications.

Additional Resources

For additional resources on logarithmic expressions, including practice problems and interactive calculators, please visit the following websites:

• Mathway: A online math problem solver that can help you with logarithmic expressions and other math topics.
• Khan Academy: A online learning platform that offers video lessons and practice problems on logarithmic expressions and other math topics.
• Wolfram Alpha: A online calculator that can help you with logarithmic expressions and other math topics.

Introduction

In our previous article, we explored how to rewrite logarithmic expressions as a single logarithm with the same base, and simplify any fractions that may arise. In this article, we will answer some of the most frequently asked questions about logarithmic expressions.

Q: What is a logarithmic expression?

A: A logarithmic expression is an expression that involves the logarithm of a number. The logarithm of a number is the power to which a base number must be raised to produce that number.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• $\log_a b - \log_a c = \log_a \frac{b}{c}$
• $\log_a b + \log_a c = \log_a (bc)$
• $\log_a b^c = c\log_a b$

Q: How do I rewrite a logarithmic expression as a single logarithm with the same base?

A: To rewrite a logarithmic expression as a single logarithm with the same base, you can use the properties of logarithms. Specifically, you can use the property that states $\log_a b - \log_a c = \log_a \frac{b}{c}$.

Q: How do I simplify a fraction in a logarithmic expression?

A: To simplify a fraction in a logarithmic expression, you can divide the numerator and denominator by their greatest common divisor.

Q: What are some real-world applications of logarithmic expressions?

A: Logarithmic expressions have a wide range of real-world applications, including:

• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic expressions are used to calculate the power of a signal and the gain of an amplifier.

Q: How do I use a calculator to evaluate a logarithmic expression?

A: To use a calculator to evaluate a logarithmic expression, you can enter the expression into the calculator and press the "log" button. The calculator will then display the result of the logarithmic expression.

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

A: Some common mistakes to avoid when working with logarithmic expressions include:

• Forgetting to change the base: Make sure to change the base of the logarithmic expression to the same base as the calculator or other tool you are using.
• Not simplifying fractions: Make sure to simplify fractions in the logarithmic expression before evaluating it.
• Not using the correct property of logarithms: Make sure to use the correct property of logarithms to rewrite the logarithmic expression as a single logarithm with the same base.

Q: How do I practice working with logarithmic expressions?

A: To practice working with logarithmic expressions, you can try the following:

• Practice problems: Try solving practice problems on logarithmic expressions, such as rewriting logarithmic expressions as a single logarithm with the same base and simplifying fractions.
• Interactive calculators: Use interactive calculators to evaluate logarithmic expressions and explore different properties of logarithms.
• Real-world applications: Try to apply logarithmic expressions to real-world problems, such as calculating interest rates or the concentration of a substance.

By following these tips and practicing working with logarithmic expressions, you can become more confident and proficient in working with these expressions.