Use Properties Of Exponents To Find The Value Of $\log _8 1$.A. 0 B. $\frac{1}{8}$ C. 8 D. 1

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Introduction

In mathematics, exponents and logarithms are two fundamental concepts that are closely related. Exponents are used to represent repeated multiplication of a number, while logarithms are used to find the power to which a base number must be raised to obtain a given value. In this article, we will explore the properties of exponents and logarithms, and use them to find the value of $\log _8 1$.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $2^3$ means $2 \times 2 \times 2$, which is equal to $8$. Exponents are used to simplify complex expressions and to represent large numbers in a more manageable form.

What are Logarithms?

Logarithms are the inverse of exponents. They are used to find the power to which a base number must be raised to obtain a given value. For example, if we want to find the value of $\log _2 8$, we need to find the power to which $2$ must be raised to obtain $8$. Since $2^3 = 8$, the value of $\log _2 8$ is $3$.

Properties of Exponents

Exponents have several important properties that are used to simplify complex expressions and to solve equations. Some of the key properties of exponents are:

  • Product of Powers: When multiplying two numbers with the same base, we can add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
  • Power of a Power: When raising a power to a power, we can multiply the exponents. For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.
  • Zero Exponent: Any number raised to the power of zero is equal to $1$. For example, $2^0 = 1$.
  • Negative Exponent: A negative exponent is equal to the reciprocal of the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Properties of Logarithms

Logarithms also have several important properties that are used to simplify complex expressions and to solve equations. Some of the key properties of logarithms are:

  • Product of Logarithms: When multiplying two numbers with the same base, we can add their logarithms. For example, $\log _2 (2^3 \times 2^4) = \log _2 2^7 = 7$.
  • Power of a Logarithm: When raising a logarithm to a power, we can multiply the exponent. For example, $(\log _2 23)4 = (\log _2 8)^4 = 4 \log _2 8 = 4 \times 3 = 12$.
  • Logarithm of 1: The logarithm of $1$ with any base is equal to $0$. For example, $\log _2 1 = 0$.

Finding the Value of $\log _8 1$

Now that we have a good understanding of the properties of exponents and logarithms, we can use them to find the value of $\log _8 1$. Since the logarithm of $1$ with any base is equal to $0$, we can conclude that $\log _8 1 = 0$.

Conclusion

In this article, we have explored the properties of exponents and logarithms, and used them to find the value of $\log _8 1$. We have seen that exponents and logarithms are closely related, and that they have several important properties that are used to simplify complex expressions and to solve equations. By understanding these properties, we can solve a wide range of mathematical problems and make sense of the world around us.

Answer

The value of $\log _8 1$ is $0$.

Final Answer

Introduction

In our previous article, we explored the properties of exponents and logarithms, and used them to find the value of $\log _8 1$. In this article, we will answer some common questions about exponents and logarithms, and provide additional examples to help you understand these important mathematical concepts.

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

A: An exponent is a shorthand way of representing repeated multiplication of a number. For example, $2^3$ means $2 \times 2 \times 2$, which is equal to $8$. A logarithm, on the other hand, is the inverse of an exponent. It is used to find the power to which a base number must be raised to obtain a given value. For example, if we want to find the value of $\log _2 8$, we need to find the power to which $2$ must be raised to obtain $8$. Since $2^3 = 8$, the value of $\log _2 8$ is $3$.

Q: What are some common properties of exponents?

A: Some common properties of exponents include:

  • Product of Powers: When multiplying two numbers with the same base, we can add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
  • Power of a Power: When raising a power to a power, we can multiply the exponents. For example, $(23)4 = 2^{3 \times 4} = 2^{12}$.
  • Zero Exponent: Any number raised to the power of zero is equal to $1$. For example, $2^0 = 1$.
  • Negative Exponent: A negative exponent is equal to the reciprocal of the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What are some common properties of logarithms?

A: Some common properties of logarithms include:

  • Product of Logarithms: When multiplying two numbers with the same base, we can add their logarithms. For example, $\log _2 (2^3 \times 2^4) = \log _2 2^7 = 7$.
  • Power of a Logarithm: When raising a logarithm to a power, we can multiply the exponent. For example, $(\log _2 23)4 = (\log _2 8)^4 = 4 \log _2 8 = 4 \times 3 = 12$.
  • Logarithm of 1: The logarithm of $1$ with any base is equal to $0$. For example, $\log _2 1 = 0$.

Q: How do I evaluate an expression with exponents and logarithms?

A: To evaluate an expression with exponents and logarithms, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate any expressions inside parentheses first.
  2. Exponents: Evaluate any exponents next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: What are some common mistakes to avoid when working with exponents and logarithms?

A: Some common mistakes to avoid when working with exponents and logarithms include:

  • Forgetting to simplify expressions: Make sure to simplify expressions with exponents and logarithms before evaluating them.
  • Confusing exponents and logarithms: Remember that exponents and logarithms are inverse operations, and that they have different properties.
  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when evaluating expressions with exponents and logarithms.

Conclusion

In this article, we have answered some common questions about exponents and logarithms, and provided additional examples to help you understand these important mathematical concepts. By following the properties of exponents and logarithms, and by using the order of operations, you can evaluate expressions with exponents and logarithms with confidence.

Final Answer

The final answer is: 0\boxed{0}