Understanding Exponential and Logarithmic Forms

Oct 13, 2025 by thesweetdesigns.com 52 views

Introduction

In mathematics, exponential and logarithmic forms are two fundamental concepts that are closely related. The exponential form represents a number as a power of another number, while the logarithmic form represents a number as the power to which another number must be raised to produce the given number. In this article, we will focus on rewriting the exponential form in logarithmic form, and we will explore the different options available.

Understanding Exponential and Logarithmic Forms

Before we dive into rewriting the exponential form in logarithmic form, let's first understand the concepts of exponential and logarithmic forms.

Exponential Form: The exponential form represents a number as a power of another number. For example, the exponential form of the number 8 is $2^3$ , where 2 is the base and 3 is the exponent.
Logarithmic Form: The logarithmic form represents a number as the power to which another number must be raised to produce the given number. For example, the logarithmic form of the number 8 is $\log_2(8)$ , where 2 is the base and 8 is the result.

Rewriting Exponential Form in Logarithmic Form

Now that we have a good understanding of exponential and logarithmic forms, let's focus on rewriting the exponential form in logarithmic form. We will use the following equation as an example:

8=2^3

To rewrite this equation in logarithmic form, we need to find the logarithm of 8 with base 2. This can be done using the following formula:

\log_b(a) = c \iff b^c = a

where $b$ is the base, $a$ is the result, and $c$ is the exponent.

Applying this formula to our example, we get:

\log_2(8) = 3

This means that the logarithm of 8 with base 2 is 3.

Analyzing the Options

Now that we have rewritten the exponential form in logarithmic form, let's analyze the options available.

Option A: $\log_8(3)=2$
Option B: $\log_3(2)=8$
Option C: $\log_2(3)=8$
Option D: $\log_2(8)=3$

Let's evaluate each option to determine which one is correct.

Option A: $\log_8(3)=2$ $lo g_{8} (3) = 2$
- This option is incorrect because the base of the logarithm is 8, but the result is 3.
Option B: $\log_3(2)=8$ $lo g_{3} (2) = 8$
- This option is incorrect because the base of the logarithm is 3, but the result is 2.
Option C: $\log_2(3)=8$ $lo g_{2} (3) = 8$
- This option is incorrect because the base of the logarithm is 2, but the result is 3.
Option D: $\log_2(8)=3$ $lo g_{2} (8) = 3$
- This option is correct because the base of the logarithm is 2, and the result is 8.

Conclusion

In conclusion, rewriting the exponential form in logarithmic form is a fundamental concept in mathematics. By understanding the relationship between exponential and logarithmic forms, we can rewrite equations in logarithmic form and analyze the options available. In this article, we have focused on rewriting the exponential form in logarithmic form and have analyzed the options available. We have determined that the correct option is $\log_2(8)=3$ .

Final Answer

The final answer is $\boxed{\log_2(8)=3}$ .

Additional Resources

For more information on rewriting exponential form in logarithmic form, please refer to the following resources:

Khan Academy: Exponents and Logarithms
Mathway: Exponents and Logarithms
Wolfram Alpha: Exponents and Logarithms

FAQs

Q: What is the difference between exponential and logarithmic forms? A: Exponential form represents a number as a power of another number, while logarithmic form represents a number as the power to which another number must be raised to produce the given number.

Q: How do I rewrite exponential form in logarithmic form? A: To rewrite exponential form in logarithmic form, use the formula $\log_b(a) = c \iff b^c = a$ , where $b$ is the base, $a$ is the result, and $c$ is the exponent.

Q&A: Frequently Asked Questions

Q: What is the correct option for rewriting the exponential form in logarithmic form? A: The correct option is $\log_2(8)=3$ .

Q: Can you provide more examples of rewriting exponential form in logarithmic form? A: Here are a few more examples:

$4=2^2 \implies \log_2(4)=2$
$9=3^2 \implies \log_3(9)=2$
$16=4^2 \implies \log_4(16)=2$

Q: How do I evaluate logarithmic expressions? A: To evaluate logarithmic expressions, use the following properties:

$\log_b(b) = 1$
$\log_b(1) = 0$
$\log_b(b^c) = c$
$\log_b(a^c) = c \log_b(a)$

Q: Can you provide more information on logarithmic properties? A: Here are a few more logarithmic properties:

$\log_b(a) + \log_b(c) = \log_b(ac)$
$\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})$
$\log_b(a) \cdot \log_b(c) = \log_b(a^{\log_b(c)})$

Q: How do I use logarithmic properties to simplify expressions? A: To use logarithmic properties to simplify expressions, follow these steps:

Identify the logarithmic properties that can be applied to the expression.
Apply the properties to simplify the expression.
Evaluate the simplified expression.

Q: Can you provide more information on logarithmic functions? A: Logarithmic functions are a type of function that is defined as the inverse of an exponential function. The logarithmic function with base $b$ is defined as:

f(x) = \log_b(x)

The domain of the logarithmic function is all positive real numbers, and the range is all real numbers.

Q: How do I graph logarithmic functions? A: To graph logarithmic functions, follow these steps:

Identify the base of the logarithmic function.
Determine the domain and range of the function.
Plot the points on the graph.
Draw the graph of the function.

Q: Can you provide more information on logarithmic graphs? A: Logarithmic graphs are a type of graph that is used to represent logarithmic functions. The graph of a logarithmic function with base $b$ is a curve that is concave down and has a vertical asymptote at $x=0$ .

Q: How do I use logarithmic graphs to solve problems? A: To use logarithmic graphs to solve problems, follow these steps:

Identify the problem that needs to be solved.
Determine the logarithmic function that is related to the problem.
Graph the logarithmic function.
Use the graph to solve the problem.