Choosing the Correct Expression

Oct 13, 2025 by miodeya.com 85 views

Introduction

In mathematics, logarithms are a fundamental concept that plays a crucial role in various branches of mathematics, including algebra, calculus, and number theory. One of the most important properties of logarithms is the power rule, which states that for any positive numbers $a$ , $b$ , and $d$ , with $b \neq 1$ , $\log_b (a^d) = d \bullet \log_b a$ . In this article, we will delve into the world of logarithms and explore this property in detail.

What are Logarithms?

Before we dive into the power rule, let's first understand what logarithms are. A logarithm is the inverse operation of exponentiation. In other words, if we have a number $x$ and we want to find the logarithm of $x$ to the base $b$ , we are essentially asking what power $b$ must be raised to in order to get $x$ . This is denoted as $\log_b x$ .

The Power Rule

Now that we have a basic understanding of logarithms, let's explore the power rule. The power rule states that for any positive numbers $a$ , $b$ , and $d$ , with $b \neq 1$ , $\log_b (a^d) = d \bullet \log_b a$ . This means that if we have a number $a$ raised to the power of $d$ , and we want to find the logarithm of this expression to the base $b$ , we can simply multiply the logarithm of $a$ to the base $b$ by $d$ .

Proof of the Power Rule

To prove the power rule, we can start by using the definition of logarithms. We know that $\log_b x = y$ if and only if $b^y = x$ . Using this definition, we can rewrite the expression $\log_b (a^d)$ as $y$ such that $b^y = a^d$ .

Step 1: Rewrite the expression

We can rewrite the expression $b^y = a^d$ as $b^y = (a^d)^1$ .

Step 2: Use the property of exponents

Using the property of exponents, we know that $(a^d)^1 = a^{d \bullet 1}$ . Therefore, we can rewrite the expression $b^y = (a^d)^1$ as $b^y = a^{d \bullet 1}$ .

Step 3: Simplify the expression

Simplifying the expression $b^y = a^{d \bullet 1}$ , we get $b^y = a^d$ .

Step 4: Take the logarithm of both sides

Taking the logarithm of both sides of the equation $b^y = a^d$ , we get $\log_b (b^y) = \log_b (a^d)$ .

Step 5: Use the property of logarithms

Using the property of logarithms, we know that $\log_b (b^y) = y$ . Therefore, we can rewrite the expression $\log_b (b^y) = \log_b (a^d)$ as $y = \log_b (a^d)$ .

Step 6: Simplify the expression

Simplifying the expression $y = \log_b (a^d)$ , we get $y = d \bullet \log_b a$ .

Conclusion

In conclusion, we have proven the power rule of logarithms, which states that for any positive numbers $a$ , $b$ , and $d$ , with $b \neq 1$ , $\log_b (a^d) = d \bullet \log_b a$ . This property is a fundamental concept in mathematics and has numerous applications in various branches of mathematics.

Choosing the Correct Expression

Now that we have proven the power rule, let's choose the correct expression from the options given. The correct expression is:

C. $\log_b a^d = d \bullet \log_b a$

The other options are incorrect because they do not accurately represent the power rule of logarithms.

Conclusion

$In this article, we have explored the power rule of logarithms, which states that for any positive numbers $a$, $b$, and $d$, with $b \neq 1$, $\log_b (a^d) = d \bullet \log_b a$. We have also proven this property using the definition of logarithms and the property of exponents. Finally, we have chosen the correct expression from the options given.<br/> **Frequently Asked Questions: Power Rule of Logarithms**$

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that for any positive numbers $a$ , $b$ , and $d$ , with $b \neq 1$ , $\log_b (a^d) = d \bullet \log_b a$ . This means that if we have a number $a$ raised to the power of $d$ , and we want to find the logarithm of this expression to the base $b$ , we can simply multiply the logarithm of $a$ to the base $b$ by $d$ .

Q: How do I apply the power rule of logarithms?

A: To apply the power rule of logarithms, you need to follow these steps:

Identify the base, $b$ , and the exponent, $d$ .
Identify the number, $a$ , that is being raised to the power of $d$ .
Take the logarithm of $a$ to the base $b$ , which is denoted as $\log_b a$ .
Multiply the logarithm of $a$ to the base $b$ by $d$ .

Q: What are some common mistakes to avoid when applying the power rule of logarithms?

A: Some common mistakes to avoid when applying the power rule of logarithms include:

Not identifying the base, $b$ , and the exponent, $d$ correctly.
Not taking the logarithm of $a$ to the base $b$ correctly.
Not multiplying the logarithm of $a$ to the base $b$ by $d$ correctly.

Q: Can I apply the power rule of logarithms to any base?

A: Yes, you can apply the power rule of logarithms to any base, as long as the base is not equal to 1. However, if the base is equal to 1, the power rule of logarithms does not apply.

Q: Can I apply the power rule of logarithms to any exponent?

A: Yes, you can apply the power rule of logarithms to any exponent, as long as the exponent is a positive number. However, if the exponent is a negative number, the power rule of logarithms does not apply.

Q: How do I use the power rule of logarithms to simplify expressions?

A: To use the power rule of logarithms to simplify expressions, you need to follow these steps:

Identify the expression that you want to simplify.
Apply the power rule of logarithms to the expression.
Simplify the resulting expression.

Q: Can I use the power rule of logarithms to solve equations?

A: Yes, you can use the power rule of logarithms to solve equations. To do this, you need to follow these steps:

Identify the equation that you want to solve.
Apply the power rule of logarithms to the equation.
Simplify the resulting equation.

Conclusion

In this article, we have answered some frequently asked questions about the power rule of logarithms. We have also provided some tips and tricks for applying the power rule of logarithms, as well as some common mistakes to avoid. By following these tips and tricks, you can become more confident and proficient in applying the power rule of logarithms to simplify expressions and solve equations.

Additional Resources

If you want to learn more about the power rule of logarithms, here are some additional resources that you can use:

Khan Academy: Logarithms
Mathway: Power Rule of Logarithms
Wolfram Alpha: Power Rule of Logarithms

By using these resources, you can gain a deeper understanding of the power rule of logarithms and how to apply it to simplify expressions and solve equations.