Which Logarithmic Equation Is Equivalent To The Exponential Equation Below? E A = 35 E^a = 35 E A = 35 A. Ln ⁡ 35 = A \ln 35 = A Ln 35 = A B. Log ⁡ A 35 = 2.5 \log_a 35 = 2.5 Lo G A ​ 35 = 2.5 C. Log ⁡ 35 2 = E \log_{35} 2 = E Lo G 35 ​ 2 = E D. Ln ⁡ A = 35 \ln A = 35 Ln A = 35

Introduction

In mathematics, logarithmic equations are used to solve exponential equations. These equations involve the use of logarithms, which are the inverse of exponential functions. In this article, we will explore how to solve logarithmic equations and provide a step-by-step guide on how to choose the correct logarithmic equation that is equivalent to a given exponential equation.

Understanding Exponential and Logarithmic Equations

Before we dive into solving logarithmic equations, it's essential to understand the relationship between exponential and logarithmic equations. An exponential equation is in the form of $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result. A logarithmic equation is in the form of $\log_a b = x$, where $a$ is the base, $b$ is the result, and $x$ is the exponent.

The Given Exponential Equation

The given exponential equation is $e^a = 35$. This equation involves the base $e$ (Euler's number) and the exponent $a$. Our goal is to find the equivalent logarithmic equation.

Step 1: Identify the Base and the Result

In the given exponential equation, the base is $e$ and the result is $35$. We need to find the logarithmic equation that has the same base and result.

Step 2: Choose the Correct Logarithmic Equation

There are four options for logarithmic equations:

A. $\ln 35 = a$ B. $\log_a 35 = 2.5$ C. $\log_{35} 2 = e$ D. $\ln a = 35$

We need to choose the correct logarithmic equation that is equivalent to the given exponential equation.

Analyzing Option A

Option A is $\ln 35 = a$. This equation involves the natural logarithm (base $e$) and the result $35$. However, the base of the logarithm is $e$, not $35$. Therefore, this option is not correct.

Analyzing Option B

Option B is $\log_a 35 = 2.5$. This equation involves the logarithm with base $a$ and the result $35$. However, the base of the logarithm is $a$, not $e$. Therefore, this option is not correct.

Analyzing Option C

Option C is $\log_{35} 2 = e$. This equation involves the logarithm with base $35$ and the result $2$. However, the base of the logarithm is $35$, not $e$, and the result is $2$, not $35$. Therefore, this option is not correct.

Analyzing Option D

Option D is $\ln a = 35$. This equation involves the natural logarithm (base $e$) and the result $35$. However, the base of the logarithm is $e$, not $a$. Therefore, this option is not correct.

Conclusion

After analyzing all the options, we can conclude that none of the options A, B, C, or D is correct. However, we can rewrite the given exponential equation as a logarithmic equation using the following steps:

1. Take the natural logarithm (base $e$) of both sides of the equation.
2. Use the property of logarithms that states $\ln a^x = x \ln a$.

Applying these steps, we get:

$\ln e^a = \ln 35$

Using the property of logarithms, we get:

$a \ln e = \ln 35$

Since $\ln e = 1$, we get:

$a = \ln 35$

Therefore, the correct logarithmic equation that is equivalent to the given exponential equation is:

$\ln 35 = a$

Final Answer

The final answer is $\boxed{A}$.

Additional Tips and Tricks

When solving logarithmic equations, it's essential to remember the following tips and tricks:

• Always identify the base and the result of the exponential equation.
• Choose the correct logarithmic equation that has the same base and result.
• Use the properties of logarithms to simplify the equation.
• Be careful when using the natural logarithm (base $e$) and the common logarithm (base $10$).

By following these tips and tricks, you can solve logarithmic equations with ease and confidence.

Conclusion

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

A: An exponential equation is in the form of $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result. A logarithmic equation is in the form of $\log_a b = x$, where $a$ is the base, $b$ is the result, and $x$ is the exponent.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to follow these steps:

1. Identify the base and the result of the equation.
2. Choose the correct logarithmic equation that has the same base and result.
3. Use the properties of logarithms to simplify the equation.
4. Solve for the exponent.

Q: What is the property of logarithms that states $\ln a^x = x \ln a$?

A: This property states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. In other words, if you have an equation of the form $\ln a^x = y$, you can rewrite it as $x \ln a = y$.

Q: How do I use the property of logarithms to simplify an equation?

A: To use the property of logarithms to simplify an equation, you need to follow these steps:

1. Identify the base and the exponent of the equation.
2. Use the property of logarithms to rewrite the equation in a simpler form.
3. Simplify the equation by combining like terms.

Q: What is the difference between the natural logarithm (base $e$) and the common logarithm (base $10$)?

A: The natural logarithm (base $e$) is a logarithm with a base of approximately $2.718$. The common logarithm (base $10$) is a logarithm with a base of $10$. The natural logarithm is used in many mathematical and scientific applications, while the common logarithm is used in many engineering and technical applications.

Q: How do I choose the correct logarithmic equation that has the same base and result?

A: To choose the correct logarithmic equation, you need to follow these steps:

1. Identify the base and the result of the equation.
2. Choose the logarithmic equation that has the same base and result.
3. Check that the equation is in the correct form (i.e., $\log_a b = x$).

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

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not identifying the base and the result of the equation.
• Not choosing the correct logarithmic equation.
• Not using the properties of logarithms to simplify the equation.
• Not solving for the exponent.

Q: How do I check my work when solving logarithmic equations?

A: To check your work when solving logarithmic equations, you need to follow these steps:

1. Plug in the values of the base and the result into the equation.
2. Simplify the equation using the properties of logarithms.
3. Check that the equation is in the correct form (i.e., $\log_a b = x$).

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

A: Logarithmic equations have many real-world applications, including:

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.
• Computer Science: Logarithmic equations are used to analyze and optimize algorithms.

Conclusion

In conclusion, logarithmic equations are a powerful tool for solving mathematical and scientific problems. By understanding the properties of logarithms and how to use them to simplify equations, you can solve logarithmic equations with ease and confidence. Remember to identify the base and the result of the equation, choose the correct logarithmic equation, and use the properties of logarithms to simplify the equation.