Given The Rule \[$\log(a \times B) = \log A + \log B\$\], Simplify \[$\log_3(27 \times 81)\$\].

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying logarithmic expressions using the rule $\log(a \times b) = \log a + \log b$. We will apply this rule to simplify the expression $\log_3(27 \times 81)$.

Understanding the Rule

The rule $\log(a \times b) = \log a + \log b$ states that the logarithm of a product of two numbers is equal to the sum of their individual logarithms. This rule is a fundamental property of logarithms and is used extensively in mathematics.

Applying the Rule

To simplify the expression $\log_3(27 \times 81)$, we can apply the rule $\log(a \times b) = \log a + \log b$. We can rewrite the expression as:

$$\log_3(27 \times 81) = \log_3(27) + \log_3(81)$$

Simplifying the Individual Logarithms

Now, we need to simplify the individual logarithms $\log_3(27)$ and $\log_3(81)$. To do this, we can use the fact that $\log_a(b) = c$ is equivalent to $a^c = b$. We can rewrite the expressions as:

$$\log_3(27) = c \implies 3^c = 27$$

$$\log_3(81) = d \implies 3^d = 81$$

Finding the Values of c and d

To find the values of c and d, we can use the fact that $3^3 = 27$ and $3^4 = 81$. Therefore, we can conclude that:

$$c = 3 \implies \log_3(27) = 3$$

$$d = 4 \implies \log_3(81) = 4$$

Substituting the Values of c and d

Now, we can substitute the values of c and d back into the expression:

$$\log_3(27 \times 81) = \log_3(27) + \log_3(81) = 3 + 4$$

Simplifying the Expression

Finally, we can simplify the expression by adding the values:

$$\log_3(27 \times 81) = 3 + 4 = 7$$

Conclusion

In this article, we have simplified the logarithmic expression $\log_3(27 \times 81)$ using the rule $\log(a \times b) = \log a + \log b$. We have applied this rule to simplify the individual logarithms and then substituted the values back into the expression. The final answer is $\boxed{7}$.

Common Mistakes to Avoid

When simplifying logarithmic expressions, it is essential to avoid common mistakes. Some of the common mistakes to avoid include:

• Not applying the rule: Failing to apply the rule $\log(a \times b) = \log a + \log b$ can lead to incorrect simplifications.
• Not simplifying individual logarithms: Failing to simplify individual logarithms can lead to incorrect results.
• Not substituting values correctly: Failing to substitute values correctly can lead to incorrect results.

Real-World Applications

Logarithmic expressions have numerous real-world applications. Some of the real-world applications of logarithmic expressions include:

• Finance: Logarithmic expressions are used in finance to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used in science to calculate the pH of a solution and the concentration of a solution.
• Engineering: Logarithmic expressions are used in engineering to calculate the power of a system and the efficiency of a system.

Final Thoughts

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will provide a Q&A guide to help you understand logarithmic expressions and how to simplify them.

Q: What is a logarithmic expression?

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse of an exponential function. Logarithmic expressions are used to solve equations and inequalities that involve exponential functions.

Q: What is the rule for simplifying logarithmic expressions?

A: The rule for simplifying logarithmic expressions is $\log(a \times b) = \log a + \log b$. This rule states that the logarithm of a product of two numbers is equal to the sum of their individual logarithms.

Q: How do I simplify a logarithmic expression using the rule?

A: To simplify a logarithmic expression using the rule, you need to follow these steps:

1. Identify the logarithmic expression: Identify the logarithmic expression that you want to simplify.
2. Apply the rule: Apply the rule $\log(a \times b) = \log a + \log b$ to the logarithmic expression.
3. Simplify individual logarithms: Simplify the individual logarithms using the fact that $\log_a(b) = c$ is equivalent to $a^c = b$.
4. Substitute values: Substitute the values of the individual logarithms back into the expression.

Q: What are some common mistakes to avoid when simplifying logarithmic expressions?

A: Some common mistakes to avoid when simplifying logarithmic expressions include:

• Not applying the rule: Failing to apply the rule $\log(a \times b) = \log a + \log b$ can lead to incorrect simplifications.
• Not simplifying individual logarithms: Failing to simplify individual logarithms can lead to incorrect results.
• Not substituting values correctly: Failing to substitute values correctly can lead to incorrect results.

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

A: Logarithmic expressions have numerous real-world applications. Some of the real-world applications of logarithmic expressions include:

• Finance: Logarithmic expressions are used in finance to calculate interest rates and investment returns.
• Science: Logarithmic expressions are used in science to calculate the pH of a solution and the concentration of a solution.
• Engineering: Logarithmic expressions are used in engineering to calculate the power of a system and the efficiency of a system.

Q: How do I evaluate logarithmic expressions with different bases?

A: To evaluate logarithmic expressions with different bases, you need to use the change of base formula, which is $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$. This formula allows you to change the base of a logarithmic expression to a different base.

Q: What is the relationship between logarithmic and exponential functions?

A: The relationship between logarithmic and exponential functions is that they are inverse functions. This means that if $f(x) = a^x$, then $f^{-1}(x) = \log_a(x)$. This relationship is essential for understanding logarithmic expressions and how to simplify them.

Q: How do I use logarithmic expressions to solve equations and inequalities?

A: To use logarithmic expressions to solve equations and inequalities, you need to follow these steps:

1. Apply the rule: Apply the rule $\log(a \times b) = \log a + \log b$ to the equation or inequality.
2. Simplify individual logarithms: Simplify the individual logarithms using the fact that $\log_a(b) = c$ is equivalent to $a^c = b$.
3. Substitute values: Substitute the values of the individual logarithms back into the equation or inequality.
4. Solve the equation or inequality: Solve the equation or inequality using algebraic techniques.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. By following the rule $\log(a \times b) = \log a + \log b$ and avoiding common mistakes, you can simplify complex logarithmic expressions and arrive at the correct answer. Remember to use logarithmic expressions to solve equations and inequalities, and to apply the change of base formula to evaluate logarithmic expressions with different bases.