Which Of These Expressions Is Equivalent To $\log (16 \cdot 14)$?A. $\log (16) + \log (14)$ B. \$\log (16) - \log (14)$[/tex\] C. $\log (16) \cdot \log (14)$ D. $16 \cdot \log (14)$

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Logarithmic expressions are a fundamental concept in mathematics, and understanding how to manipulate them is crucial for solving various mathematical problems. In this article, we will explore the concept of logarithmic expressions and determine which of the given expressions is equivalent to $\log (16 \cdot 14)$.

What are Logarithmic Expressions?

A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to produce a given value. In other words, it is the inverse operation of exponentiation. The general form of a logarithmic expression is:

log⁡b(x)=y\log_b(x) = y

where bb is the base, xx is the value, and yy is the exponent.

Properties of Logarithmic Expressions

There are several properties of logarithmic expressions that are essential to understand. These properties include:

  • Product Rule: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)
  • Quotient Rule: log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)
  • Power Rule: log⁡b(xy)=y⋅log⁡b(x)\log_b(x^y) = y \cdot \log_b(x)

Evaluating the Given Expression

We are given the expression $\log (16 \cdot 14)$. Using the product rule, we can rewrite this expression as:

log⁡(16⋅14)=log⁡(16)+log⁡(14)\log (16 \cdot 14) = \log (16) + \log (14)

This is because the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Comparing the Options

Now that we have evaluated the given expression, let's compare it with the options provided:

A. $\log (16) + \log (14)$ B. $\log (16) - \log (14)$ C. $\log (16) \cdot \log (14)$ D. $16 \cdot \log (14)$

Based on our evaluation, we can see that option A is equivalent to the given expression $\log (16 \cdot 14)$.

Conclusion

In conclusion, the expression $\log (16 \cdot 14)$ is equivalent to $\log (16) + \log (14)$. This is because of the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Understanding logarithmic expressions and their properties is crucial for solving various mathematical problems, and this article has provided a comprehensive overview of the concept.

Additional Examples

To further illustrate the concept, let's consider a few additional examples:

  • log⁡(24⋅36)=log⁡(24)+log⁡(36)\log (24 \cdot 36) = \log (24) + \log (36)

  • log⁡(4816)=log⁡(48)−log⁡(16)\log (\frac{48}{16}) = \log (48) - \log (16)

  • log⁡(24⋅32)=4⋅log⁡(2)+2⋅log⁡(3)\log (2^4 \cdot 3^2) = 4 \cdot \log (2) + 2 \cdot \log (3)

These examples demonstrate the application of the product rule, quotient rule, and power rule in evaluating logarithmic expressions.

Final Thoughts

In the previous article, we explored the concept of logarithmic expressions and determined which of the given expressions is equivalent to $\log (16 \cdot 14)$. In this article, we will answer some frequently asked questions about logarithmic expressions.

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

A: A logarithmic expression is the inverse operation of an exponential expression. While an exponential expression represents the power to which a base number must be raised to produce a given value, a logarithmic expression represents the exponent to which a base number must be raised to produce a given value.

Q: What is the product rule for logarithmic expressions?

A: The product rule for logarithmic expressions states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In other words:

log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)

Q: What is the quotient rule for logarithmic expressions?

A: The quotient rule for logarithmic expressions states that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors. In other words:

log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)

Q: What is the power rule for logarithmic expressions?

A: The power rule for logarithmic expressions states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. In other words:

log⁡b(xy)=y⋅log⁡b(x)\log_b(x^y) = y \cdot \log_b(x)

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the product rule, quotient rule, and power rule to simplify the expression. For example, to evaluate $\log (16 \cdot 14)$, you can use the product rule to rewrite the expression as $\log (16) + \log (14)$.

Q: What is the base of a logarithmic expression?

A: The base of a logarithmic expression is the number to which the logarithm is being taken. For example, in the expression $\log_2(8)$, the base is 2.

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

A: Yes, you can use a calculator to evaluate a logarithmic expression. Most calculators have a logarithm function that allows you to enter the base and the value, and it will return the logarithm of the value.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

  • log⁡b(x)\log_b(x)

  • log⁡b(xy)\log_b(xy)

  • log⁡b(xy)\log_b(\frac{x}{y})

  • log⁡b(xy)\log_b(x^y)

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a graphing software. You can also use a table of values to create a graph.

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

A: Logarithmic expressions have many 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 frequency of a wave.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding how to evaluate and manipulate them is crucial for solving various mathematical problems. This article has provided a comprehensive overview of the concept, including the product rule, quotient rule, and power rule, as well as some frequently asked questions and answers.