Use The Laws Of Logarithms To Evaluate The Expression:$\log _2\left(8^{83}\right$\]

Understanding the Laws of Logarithms

The laws of logarithms are a set of rules that help us simplify and evaluate logarithmic expressions. These rules are essential in mathematics, particularly in algebra and calculus. In this article, we will focus on using the laws of logarithms to evaluate the expression $\log _2\left(8^{83}\right)$.

The Power Rule of Logarithms

One of the most important laws of logarithms is the power rule, which states that $\log _a\left(x^y\right) = y\log _a\left(x\right)$. This rule allows us to simplify expressions like $\log _2\left(8^{83}\right)$ by using the power rule.

Applying the Power Rule

To evaluate the expression $\log _2\left(8^{83}\right)$, we can use the power rule. We know that $8 = 2^3$, so we can rewrite the expression as $\log _2\left(\left(2^3\right)^{83}\right)$. Now, we can apply the power rule to simplify the expression.

Simplifying the Expression

Using the power rule, we can simplify the expression as follows:

$\log _2\left(\left(2^3\right)^{83}\right) = 83\log _2\left(2^3\right)$

Evaluating the Logarithm

Now, we can evaluate the logarithm using the property of logarithms that states $\log _a\left(a\right) = 1$. In this case, we have $\log _2\left(2^3\right) = 3$, since $2^3 = 8$.

Final Answer

Therefore, the final answer to the expression $\log _2\left(8^{83}\right)$ is $83 \times 3 = 249$.

Real-World Applications

The laws of logarithms have numerous real-world applications in fields such as engineering, economics, and computer science. For example, logarithmic scales are used to measure sound levels, earthquake intensities, and financial returns. In addition, logarithmic functions are used to model population growth, chemical reactions, and electrical circuits.

Conclusion

In conclusion, the laws of logarithms are a powerful tool for simplifying and evaluating logarithmic expressions. By applying the power rule and other laws of logarithms, we can simplify complex expressions and arrive at a final answer. The laws of logarithms have numerous real-world applications and are an essential part of mathematics.

Additional Examples

Here are a few additional examples of using the laws of logarithms to evaluate expressions:

• $\log _3\left(9^{42}\right) = 42\log _3\left(3^2\right) = 42 \times 2 = 84$
• $\log _5\left(25^{31}\right) = 31\log _5\left(5^2\right) = 31 \times 2 = 62$
• $\log _7\left(49^{29}\right) = 29\log _7\left(7^2\right) = 29 \times 2 = 58$

Practice Problems

Here are a few practice problems to help you apply the laws of logarithms:

• Evaluate the expression $\log _4\left(16^{67}\right)$.
• Evaluate the expression $\log _6\left(36^{41}\right)$.
• Evaluate the expression $\log _8\left(64^{33}\right)$.

Solutions

Here are the solutions to the practice problems:

• $\log _4\left(16^{67}\right) = 67\log _4\left(4^2\right) = 67 \times 2 = 134$
• $\log _6\left(36^{41}\right) = 41\log _6\left(6^2\right) = 41 \times 2 = 82$
• $\log _8\left(64^{33}\right) = 33\log _8\left(8^2\right) = 33 \times 2 = 66$

Conclusion

In conclusion, the laws of logarithms are a powerful tool for simplifying and evaluating logarithmic expressions. By applying the power rule and other laws of logarithms, we can simplify complex expressions and arrive at a final answer. The laws of logarithms have numerous real-world applications and are an essential part of mathematics.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\log _a\left(x^y\right) = y\log _a\left(x\right)$. This rule allows us to simplify expressions like $\log _2\left(8^{83}\right)$ by using the power rule.

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, we need to rewrite the expression in the form $\log _a\left(x^y\right)$. Then, we can use the power rule to simplify the expression as $y\log _a\left(x\right)$.

Q: What is the property of logarithms that states $\log _a\left(a\right) = 1$?

A: This property is known as the identity property of logarithms. It states that the logarithm of a number to its own base is equal to 1.

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

A: To evaluate a logarithmic expression using the power rule, we need to follow these steps:

1. Rewrite the expression in the form $\log _a\left(x^y\right)$.
2. Apply the power rule to simplify the expression as $y\log _a\left(x\right)$.
3. Evaluate the logarithm using the property of logarithms that states $\log _a\left(a\right) = 1$.

Q: What are some real-world applications of the laws of logarithms?

A: The laws of logarithms have numerous real-world applications in fields such as engineering, economics, and computer science. For example, logarithmic scales are used to measure sound levels, earthquake intensities, and financial returns. In addition, logarithmic functions are used to model population growth, chemical reactions, and electrical circuits.

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

A: The product rule states that $\log _a\left(xy\right) = \log _a\left(x\right) + \log _a\left(y\right)$. To simplify a logarithmic expression using the product rule, we need to follow these steps:

1. Rewrite the expression in the form $\log _a\left(xy\right)$.
2. Apply the product rule to simplify the expression as $\log _a\left(x\right) + \log _a\left(y\right)$.

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

A: The quotient rule states that $\log _a\left(\frac{x}{y}\right) = \log _a\left(x\right) - \log _a\left(y\right)$. To simplify a logarithmic expression using the quotient rule, we need to follow these steps:

1. Rewrite the expression in the form $\log _a\left(\frac{x}{y}\right)$.
2. Apply the quotient rule to simplify the expression as $\log _a\left(x\right) - \log _a\left(y\right)$.

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

A: Some common mistakes to avoid when working with logarithmic expressions include:

• Forgetting to apply the power rule when simplifying expressions.
• Not using the correct base when evaluating logarithms.
• Not following the order of operations when simplifying expressions.

Q: How do I check my work when simplifying logarithmic expressions?

A: To check your work when simplifying logarithmic expressions, you can follow these steps:

1. Rewrite the original expression.
2. Simplify the expression using the laws of logarithms.
3. Check that the simplified expression is equivalent to the original expression.

Q: What are some additional resources for learning about logarithmic expressions?

A: Some additional resources for learning about logarithmic expressions include:

• Online tutorials and videos.
• Textbooks and workbooks.
• Online communities and forums.
• Practice problems and exercises.

Conclusion

In conclusion, the laws of logarithms are a powerful tool for simplifying and evaluating logarithmic expressions. By applying the power rule, product rule, and quotient rule, we can simplify complex expressions and arrive at a final answer. The laws of logarithms have numerous real-world applications and are an essential part of mathematics.