Use The Properties Of Logarithms To Evaluate Each Of The Following:(a) Log ⁡ 3 48 − 2 Log ⁡ 3 4 = □ \log_3 48 - 2\log_3 4 = \square Lo G 3 ​ 48 − 2 Lo G 3 ​ 4 = □ (b) Ln ⁡ E 4 + Ln ⁡ E − 7 = □ \ln E^4 + \ln E^{-7} = \square Ln E 4 + Ln E − 7 = □

Introduction

Logarithms are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and finance. In this article, we will explore the properties of logarithms and learn how to evaluate logarithmic expressions using these properties.

Properties of Logarithms

Before we dive into evaluating logarithmic expressions, let's review the properties of logarithms. The properties of logarithms are as follows:

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

Evaluating Logarithmic Expressions

Now that we have reviewed the properties of logarithms, let's evaluate the given logarithmic expressions.

(a) $\log_3 48 - 2\log_3 4 = \square$

To evaluate this expression, we can use the quotient property and the power property of logarithms.

$\log_3 48 - 2\log_3 4 = \log_3 \left(\frac{48}{4^2}\right)$

Using the quotient property, we can rewrite the expression as:

$\log_3 \left(\frac{48}{16}\right)$

Simplifying the expression, we get:

$\log_3 3$

Using the power property, we can rewrite the expression as:

$\log_3 3^1$

Simplifying the expression, we get:

$1$

Therefore, the value of the expression is:

$\boxed{1}$

(b) $\ln e^4 + \ln e^{-7} = \square$

To evaluate this expression, we can use the product property and the power property of logarithms.

$\ln e^4 + \ln e^{-7} = \ln (e^4 \cdot e^{-7})$

Using the product property, we can rewrite the expression as:

$\ln e^{4-7}$

Simplifying the expression, we get:

$\ln e^{-3}$

Using the power property, we can rewrite the expression as:

$-3\ln e$

Simplifying the expression, we get:

$-3$

Therefore, the value of the expression is:

$\boxed{-3}$

Conclusion

In this article, we have learned how to evaluate logarithmic expressions using the properties of logarithms. We have used the product property, quotient property, power property, and change of base property to simplify the given expressions. By applying these properties, we have been able to evaluate the expressions and find their values.

Applications of Logarithmic Expressions

Logarithmic expressions have numerous applications in various fields, including science, engineering, and finance. Some of the applications of logarithmic expressions include:

• Sound Level Measurement: Logarithmic expressions are used to measure sound levels in decibels.
• pH Measurement: Logarithmic expressions are used to measure pH levels in chemistry.
• Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
• Computer Science: Logarithmic expressions are used in algorithms and data structures.

Real-World Examples

Here are some real-world examples of logarithmic expressions:

• Sound Level Measurement: A sound level of 80 decibels is equivalent to $10^{\frac{80}{10}}$ or $10^8$.
• pH Measurement: A pH level of 7 is equivalent to $\log_{10} 10^7$ or 7.
• Finance: An interest rate of 5% is equivalent to $\log_{1.05} 1.05$ or 1.
• Computer Science: A binary search algorithm uses logarithmic expressions to find the location of a target value in a sorted array.

Final Thoughts

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the properties of logarithms and how to evaluate logarithmic expressions, we can solve a wide range of problems and make informed decisions in our personal and professional lives.

References

• "Logarithms" by Math Is Fun
• "Properties of Logarithms" by Khan Academy
• "Logarithmic Expressions" by Wolfram MathWorld

Glossary

• Logarithm: The inverse operation of exponentiation.
• Base: The number that is raised to a power in an exponential expression.
• Exponent: The power to which a base is raised in an exponential expression.
• Product Property: A property of logarithms that states $\log_b (xy) = \log_b x + \log_b y$.
• Quotient Property: A property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$.
• Power Property: A property of logarithms that states $\log_b x^y = y\log_b x$.
• Change of Base Property: A property of logarithms that states $\log_b x = \frac{\log_a x}{\log_a b}$.

Further Reading

If you want to learn more about logarithmic expressions and their applications, here are some recommended resources:

• "Logarithms and Exponents" by MIT OpenCourseWare
• "Logarithmic Functions" by Wolfram MathWorld
• "Logarithmic Expressions" by Khan Academy

I hope this article has been helpful in understanding the properties of logarithms and how to evaluate logarithmic expressions. If you have any questions or need further clarification, please don't hesitate to ask.

Introduction

In our previous article, we explored the properties of logarithms and learned how to evaluate logarithmic expressions using these properties. In this article, we will answer some frequently asked questions about logarithmic expressions and provide additional examples to help solidify your understanding.

Q&A

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 $y = \log_a x$. Exponents and logarithms are related but distinct concepts.

Q: How do I evaluate a logarithmic expression with a negative exponent?

A: To evaluate a logarithmic expression with a negative exponent, you can use the quotient property of logarithms. For example, $\log_b \left(\frac{1}{x}\right) = -\log_b x$.

Q: Can I use a calculator to evaluate logarithmic expressions?

A: Yes, you can use a calculator to evaluate logarithmic expressions. Most calculators have a logarithm function that allows you to input a base and a value and calculate the logarithm.

Q: How do I change the base of a logarithmic expression?

A: To change the base of a logarithmic expression, you can use the change of base property of logarithms. For example, $\log_b x = \frac{\log_a x}{\log_a b}$.

Q: Can I use logarithmic expressions to solve equations?

A: Yes, you can use logarithmic expressions to solve equations. For example, if you have an equation of the form $x^y = z$, you can take the logarithm of both sides to solve for $x$.

Q: How do I evaluate a logarithmic expression with a fractional exponent?

A: To evaluate a logarithmic expression with a fractional exponent, you can use the power property of logarithms. For example, $\log_b x^{\frac{1}{2}} = \frac{1}{2} \log_b x$.

Q: Can I use logarithmic expressions to model real-world phenomena?

A: Yes, you can use logarithmic expressions to model real-world phenomena. For example, the growth of a population can be modeled using a logarithmic function.

Additional Examples

Here are some additional examples of logarithmic expressions:

• Example 1: Evaluate $\log_2 16$.
• Solution: Using the power property of logarithms, we have $\log_2 16 = \log_2 2^4 = 4$.
• Example 2: Evaluate $\log_5 \left(\frac{1}{25}\right)$.
• Solution: Using the quotient property of logarithms, we have $\log_5 \left(\frac{1}{25}\right) = -\log_5 25 = -2$.
• Example 3: Evaluate $\log_3 \left(\sqrt{9}\right)$.
• Solution: Using the power property of logarithms, we have $\log_3 \left(\sqrt{9}\right) = \log_3 3^{\frac{1}{2}} = \frac{1}{2}$.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic expressions and provided additional examples to help solidify your understanding. We hope this article has been helpful in your study of logarithmic expressions.

Glossary

• Logarithm: The inverse operation of exponentiation.
• Base: The number that is raised to a power in an exponential expression.
• Exponent: The power to which a base is raised in an exponential expression.
• Product Property: A property of logarithms that states $\log_b (xy) = \log_b x + \log_b y$.
• Quotient Property: A property of logarithms that states $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$.
• Power Property: A property of logarithms that states $\log_b x^y = y\log_b x$.
• Change of Base Property: A property of logarithms that states $\log_b x = \frac{\log_a x}{\log_a b}$.

Further Reading

If you want to learn more about logarithmic expressions and their applications, here are some recommended resources:

• "Logarithms and Exponents" by MIT OpenCourseWare
• "Logarithmic Functions" by Wolfram MathWorld
• "Logarithmic Expressions" by Khan Academy

I hope this article has been helpful in your study of logarithmic expressions. If you have any questions or need further clarification, please don't hesitate to ask.