Which Expressions Are Equivalent To The One Below? Check All That Apply. Log ⁡ 2 2 + Log ⁡ 2 8 \log _2 2+\log _2 8 Lo G 2 ​ 2 + Lo G 2 ​ 8 A. 4 B. \log _2\left(2^4\right ] C. Log ⁡ 10 \log 10 Lo G 10 D. Log ⁡ 2 16 \log _2 16 Lo G 2 ​ 16

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will explore the given expression $\log _2 2+\log _2 8$ and determine which of the provided options are equivalent to it.

The Given Expression

The given expression is $\log _2 2+\log _2 8$. This expression involves the sum of two logarithmic terms with the same base, 2. To simplify this expression, we can use the property of logarithms that states $\log _a b + \log _a c = \log _a (b \cdot c)$.

Simplifying the Expression

Using the property mentioned above, we can rewrite the given expression as:

$$\log _2 2+\log _2 8 = \log _2 (2 \cdot 8) = \log _2 16$$

This simplification is based on the fact that the logarithm of a product is equal to the sum of the logarithms of the individual terms.

Evaluating the Options

Now that we have simplified the given expression, let's evaluate the options provided:

A. 4

This option is not equivalent to the simplified expression. The value 4 is a numerical constant, whereas the simplified expression is a logarithmic expression.

B. $\log _2\left(2^4\right)$

This option is equivalent to the simplified expression. Using the property of logarithms that states $\log _a a^b = b$, we can rewrite the expression as:

$$\log _2 16 = \log _2\left(2^4\right) = 4$$

This shows that option B is indeed equivalent to the simplified expression.

C. $\log 10$

This option is not equivalent to the simplified expression. The expression $\log 10$ is a logarithmic expression with base 10, whereas the simplified expression has base 2.

D. $\log _2 16$

This option is equivalent to the simplified expression. As we have already shown, the simplified expression is $\log _2 16$, which is identical to option D.

Conclusion

In conclusion, the options that are equivalent to the given expression $\log _2 2+\log _2 8$ are:

• $\log _2\left(2^4\right)$
• $\log _2 16$

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will provide a comprehensive Q&A guide to help you better understand 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. For example, the logarithm of 100 with base 10 is 2, because 10^2 = 100.

Q: What are the properties of logarithmic expressions?

A: The properties of logarithmic expressions are:

• Product Rule: $\log _a (b \cdot c) = \log _a b + \log _a c$
• Quotient Rule: $\log _a \left(\frac{b}{c}\right) = \log _a b - \log _a c$
• Power Rule: $\log _a b^c = c \cdot \log _a b$
• Change of Base Rule: $\log _a b = \frac{\log _c b}{\log _c a}$

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithmic expressions. For example, if you have the expression $\log _2 2+\log _2 8$, you can use the product rule to simplify it as:

$$\log _2 2+\log _2 8 = \log _2 (2 \cdot 8) = \log _2 16$$

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

A: A logarithmic expression is an expression that involves the logarithm of a number, whereas an exponential expression is an expression that involves the exponentiation of a number. For example, the expression $\log _2 16$ is a logarithmic expression, whereas the expression $2^4$ is an exponential expression.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the expression. For example, if you have the expression $\log _2 16$, you can evaluate it as:

$$\log _2 16 = 4$$

because $2^4 = 16$.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log _2 2$
• $\log _2 4$
• $\log _2 8$
• $\log _2 16$
• $\log _2 32$
• $\log _2 64$

Q: How do I use logarithmic expressions in real-world problems?

A: Logarithmic expressions are used in a variety of real-world problems, 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 frequency of a wave.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. By using the properties of logarithmic expressions and evaluating them correctly, you can solve a wide range of problems in finance, science, and engineering.