Evaluate $\log _5(1.2$\].A. 0.113 B. -0.031 C. -0.216 D. 0.257

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will focus on evaluating the logarithmic expression $\log _5(1.2)$ and provide a step-by-step guide on how to solve it.

Understanding Logarithms

Before we dive into the evaluation of the logarithmic expression, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log _a(c) = b$. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number.

Evaluating $\log _5(1.2)$

To evaluate $\log _5(1.2)$, we need to find the exponent to which 5 must be raised to produce 1.2. This can be done using a calculator or by using the change of base formula.

Method 1: Using a Calculator

One way to evaluate $\log _5(1.2)$ is to use a calculator. Most calculators have a built-in logarithm function that can be used to evaluate logarithmic expressions.

• Step 1: Enter the expression $\log _5(1.2)$ into the calculator.
• Step 2: Press the "log" or "ln" button to evaluate the expression.
• Step 3: The calculator will display the result of the evaluation.

Method 2: Using the Change of Base Formula

Another way to evaluate $\log _5(1.2)$ is to use the change of base formula. The change of base formula states that $\log _a(b) = \frac{\log _c(b)}{\log _c(a)}$, where $c$ is any positive real number.

• Step 1: Choose a base $c$ (e.g., 10 or $e$).
• Step 2: Evaluate $\log _c(1.2)$ and $\log _c(5)$.
• Step 3: Substitute the values into the change of base formula and simplify.

Solving the Expression

Using a calculator, we find that $\log _5(1.2) \approx 0.113$. This is the correct answer.

Conclusion

Evaluating logarithmic expressions is an essential skill in mathematics. By understanding the concept of logarithms and using the change of base formula or a calculator, we can evaluate expressions like $\log _5(1.2)$. In this article, we provided a step-by-step guide on how to evaluate this expression and discussed the importance of logarithmic expressions in mathematics.

Frequently Asked Questions

• Q: What is the logarithm of 1.2 to the base 5? A: The logarithm of 1.2 to the base 5 is approximately 0.113.
• Q: How do I evaluate a logarithmic expression? A: You can use a calculator or the change of base formula to evaluate a logarithmic expression.
• Q: What is the change of base formula? A: The change of base formula states that $\log _a(b) = \frac{\log _c(b)}{\log _c(a)}$, where $c$ is any positive real number.

References

• [1] "Logarithms." MathWorld, Wolfram Research.
• [2] "Change of Base Formula." MathWorld, Wolfram Research.

Additional Resources

• [1] Khan Academy. "Logarithms." Khan Academy, 2022.
• [2] MIT OpenCourseWare. "18.01 Single Variable Calculus." MIT OpenCourseWare, 2022.

About the Author

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for students and professionals alike. In this article, we will provide a Q&A guide on logarithmic expressions, covering various topics and concepts.

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log _a(c) = b$. 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 is the change of base formula?

A: The change of base formula states that $\log _a(b) = \frac{\log _c(b)}{\log _c(a)}$, where $c$ is any positive real number. This formula allows us to change the base of a logarithmic expression from one base to another.

Q: How do I evaluate a logarithmic expression?

A: You can use a calculator or the change of base formula to evaluate a logarithmic expression. If you are using a calculator, simply enter the expression and press the "log" or "ln" button. If you are using the change of base formula, choose a base $c$, evaluate $\log _c(b)$ and $\log _c(a)$, and substitute the values into the formula.

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

A: A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log _a(c) = b$. An exponent, on the other hand, is the power to which a number is raised. For example, $2^3 = 8$, where 3 is the exponent.

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 built-in logarithm function that can be used to evaluate logarithmic expressions.

Q: What is the logarithm of 1 to any base?

A: The logarithm of 1 to any base is 0. This is because $a^0 = 1$ for any positive real number $a$.

Q: What is the logarithm of 0 to any base?

A: The logarithm of 0 to any base is undefined. This is because there is no positive real number $a$ such that $a^b = 0$ for any exponent $b$.

Q: Can I use the change of base formula to evaluate logarithmic expressions with negative bases?

A: No, you cannot use the change of base formula to evaluate logarithmic expressions with negative bases. The change of base formula only works for positive real numbers.

Q: What is the logarithm of a negative number to any base?

A: The logarithm of a negative number to any base is undefined. This is because there is no positive real number $a$ such that $a^b = -c$ for any exponent $b$ and any positive real number $c$.

Conclusion

Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for students and professionals alike. In this article, we provided a Q&A guide on logarithmic expressions, covering various topics and concepts. We hope this guide has been helpful in understanding logarithmic expressions and their applications.

Frequently Asked Questions

• Q: What is a logarithm? A: A logarithm is the inverse operation of exponentiation.
• Q: What is the change of base formula? A: The change of base formula states that $\log _a(b) = \frac{\log _c(b)}{\log _c(a)}$, where $c$ is any positive real number.
• Q: How do I evaluate a logarithmic expression? A: You can use a calculator or the change of base formula to evaluate a logarithmic expression.

References

• [1] "Logarithms." MathWorld, Wolfram Research.
• [2] "Change of Base Formula." MathWorld, Wolfram Research.

Additional Resources

• [1] Khan Academy. "Logarithms." Khan Academy, 2022.
• [2] MIT OpenCourseWare. "18.01 Single Variable Calculus." MIT OpenCourseWare, 2022.