Which Logarithmic Equation Is Equivalent To The Exponential Equation Below?$3^c = 27$A. $\log_{27} C = 3$ B. $\log_c 27 = 3$ C. $\log_3 C = 27$ D. $\log_3 27 = C$

Feb 27, 2025 by ADMIN 168 views

**Solving Logarithmic Equations: A Step-by-Step Guide**

Introduction

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will explore the concept of logarithmic equations and learn how to solve them. We will also focus on finding the equivalent logarithmic equation for a given exponential equation.

Understanding Exponential and Logarithmic Equations

Before we dive into solving logarithmic equations, let's first understand the concept of exponential and logarithmic equations.

An exponential equation is an equation that involves an exponent, which is a number or expression raised to a power. For example, the equation $3^c = 27$ is an exponential equation, where $3$ is the base and $c$ is the exponent.

A logarithmic equation, on the other hand, is an equation that involves a logarithm, which is the inverse operation of exponentiation. The logarithm of a number is the exponent to which a base must be raised to produce that number. For example, the equation $\log_{27} c = 3$ is a logarithmic equation, where $27$ is the base and $c$ is the argument.

Converting Exponential Equations to Logarithmic Equations

Now that we have a basic understanding of exponential and logarithmic equations, let's learn how to convert an exponential equation to a logarithmic equation.

To convert an exponential equation to a logarithmic equation, we need to use the following formula:

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

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

Using this formula, we can convert the exponential equation $3^c = 27$ to a logarithmic equation.

Step 1: Identify the Base and Argument

In the exponential equation $3^c = 27$ , the base is $3$ and the argument is $27$ .

Step 2: Use the Formula to Convert the Equation

Using the formula $\log_b a = c \iff b^c = a$ , we can convert the exponential equation to a logarithmic equation.

\log_{3} 27 = c

Step 3: Simplify the Equation (Optional)

In this case, we can simplify the equation by using the fact that $27 = 3^3$ .

\log_{3} 3^3 = c

Using the property of logarithms that $\log_b b^x = x$ , we can simplify the equation further.

3 = c

However, this is not the correct answer. We need to find the equivalent logarithmic equation for the given exponential equation.

Step 4: Find the Equivalent Logarithmic Equation

Using the formula $\log_b a = c \iff b^c = a$ , we can find the equivalent logarithmic equation.

\log_{3} 27 = c \iff 3^c = 27

This is the correct answer.

Conclusion

In this article, we learned how to solve logarithmic equations and find the equivalent logarithmic equation for a given exponential equation. We used the formula $\log_b a = c \iff b^c = a$ to convert the exponential equation $3^c = 27$ to a logarithmic equation.

Answer

The correct answer is:

D. $\log_3 27 = c$

Discussion

This problem requires a basic understanding of exponential and logarithmic equations. The student needs to be able to convert an exponential equation to a logarithmic equation using the formula $\log_b a = c \iff b^c = a$ .

Tips and Tricks

Make sure to identify the base and argument in the exponential equation.
Use the formula $\log_b a = c \iff b^c = a$ to convert the exponential equation to a logarithmic equation.
Simplify the equation if possible.
Find the equivalent logarithmic equation using the formula $\log_b a = c \iff b^c = a$ .

Practice Problems

Convert the exponential equation $2^x = 8$ to a logarithmic equation.
Find the equivalent logarithmic equation for the exponential equation $4^y = 256$ .
Convert the exponential equation $5^z = 3125$ to a logarithmic equation.

Answer Key

$\log_2 8 = x$
$\log_4 256 = y$
$\log_5 3125 = z$
Logarithmic Equations: A Q&A Guide =====================================

Introduction

Logarithmic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and computer science. In this article, we will provide a comprehensive Q&A guide to logarithmic equations, covering various topics and concepts.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The logarithm of a number is the exponent to which a base must be raised to produce that number.

Q: How do I convert an exponential equation to a logarithmic equation?

A: To convert an exponential equation to a logarithmic equation, you need to use the following formula:

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

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

Q: What is the base in a logarithmic equation?

A: The base in a logarithmic equation is the number that is raised to a power to produce the argument. For example, in the equation $\log_3 27 = c$ , the base is $3$ .

Q: What is the argument in a logarithmic equation?

A: The argument in a logarithmic equation is the number that is the result of raising the base to a power. For example, in the equation $\log_3 27 = c$ , the argument is $27$ .

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the variable (the exponent) using the properties of logarithms. For example, to solve the equation $\log_3 27 = c$ , you can use the fact that $27 = 3^3$ to simplify the equation.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

$\log_b b^x = x$
$\log_b 1 = 0$
$\log_b b = 1$
$\log_b (m^n) = n \log_b m$
$\log_b (m/n) = \log_b m - \log_b n$

Q: How do I use the properties of logarithms to solve a logarithmic equation?

A: To use the properties of logarithms to solve a logarithmic equation, you need to apply the properties in the correct order. For example, to solve the equation $\log_3 27 = c$ , you can use the fact that $27 = 3^3$ to simplify the equation.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation $\log_3 27 = c$ is a logarithmic equation, while the equation $3^c = 27$ is an exponential equation.

Q: How do I convert a logarithmic equation to an exponential equation?

A: To convert a logarithmic equation to an exponential equation, you need to use the following formula:

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

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

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

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

Not identifying the base and argument correctly
Not using the properties of logarithms correctly
Not simplifying the equation correctly
Not checking the domain and range of the logarithmic function

Conclusion

In this article, we provided a comprehensive Q&A guide to logarithmic equations, covering various topics and concepts. We hope that this guide has been helpful in understanding logarithmic equations and how to solve them.

Practice Problems

Convert the exponential equation $2^x = 8$ to a logarithmic equation.
Find the equivalent logarithmic equation for the exponential equation $4^y = 256$ .
Convert the exponential equation $5^z = 3125$ to a logarithmic equation.

Answer Key

$\log_2 8 = x$
$\log_4 256 = y$
$\log_5 3125 = z$

Discussion

Tips and Tricks

Make sure to identify the base and argument in the exponential equation.
Use the formula $\log_b a = c \iff b^c = a$ to convert the exponential equation to a logarithmic equation.
Simplify the equation if possible.
Find the equivalent logarithmic equation using the formula $\log_b a = c \iff b^c = a$ .

Additional Resources

Khan Academy: Logarithmic Equations
Mathway: Logarithmic Equations
Wolfram Alpha: Logarithmic Equations