Which Logarithmic Equation Is Equivalent To The Exponential Equation Below?$e^a = 28.37$A. $\log_{28.37} 3.65 = E$B. $\log_a 28.37 = 3.65$C. $\ln 28.37 = A$D. $\ln A = 28.37$

Introduction

Logarithmic equations and exponential equations are two fundamental concepts in mathematics that are closely related. In this article, we will explore the relationship between these two types of equations and learn how to solve logarithmic equations that are equivalent to exponential equations.

Understanding Exponential Equations

An exponential equation is an equation that involves an exponential function, which is a function that raises a base number to a power. The general form of an exponential equation is:

a^x = b

where a is the base number, x is the exponent, and b is the result.

In the given problem, we have the exponential equation:

e^a = 28.37

This equation states that the base number e (Euler's number) raised to the power of a equals 28.37.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The general form of a logarithmic equation is:

log_a b = x

where a is the base number, b is the result, and x is the exponent.

In the given problem, we have four options for logarithmic equations that are equivalent to the exponential equation e^a = 28.37. Let's analyze each option:

Option A: $\log_{28.37} 3.65 = e$

This option involves a logarithmic function with a base of 28.37 and a result of 3.65. However, the equation is not equivalent to the exponential equation e^a = 28.37, as the base and result are not the same.

Option B: $\log_a 28.37 = 3.65$

This option involves a logarithmic function with a base of a and a result of 28.37. However, the equation is not equivalent to the exponential equation e^a = 28.37, as the base is not specified.

Option C: $\ln 28.37 = a$

This option involves a natural logarithmic function with a base of e and a result of 28.37. However, the equation is not equivalent to the exponential equation e^a = 28.37, as the base is not the same.

Option D: $\ln a = 28.37$

This option involves a natural logarithmic function with a base of e and a result of a. However, the equation is not equivalent to the exponential equation e^a = 28.37, as the base is not the same.

Solving the Problem

To solve the problem, we need to find the logarithmic equation that is equivalent to the exponential equation e^a = 28.37. We can start by taking the natural logarithm of both sides of the equation:

ln(e^a) = ln(28.37)

Using the property of logarithms that states ln(a^b) = b * ln(a), we can simplify the equation:

a * ln(e) = ln(28.37)

Since ln(e) = 1, we can simplify the equation further:

a = ln(28.37)

This is the logarithmic equation that is equivalent to the exponential equation e^a = 28.37.

Conclusion

In conclusion, the logarithmic equation that is equivalent to the exponential equation e^a = 28.37 is:

a = ln(28.37)

This equation states that the base number a is equal to the natural logarithm of 28.37.

Final Answer

The final answer is:

C. $\ln 28.37 = a$

Introduction

Logarithmic equations can be a challenging topic for many students. In this article, we will provide a Q&A guide to help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. The general form of a logarithmic equation is:

log_a b = x

where a is the base number, b is the result, and x is the exponent.

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

A: A logarithmic equation is the inverse of an exponential equation. While an exponential equation raises a base number to a power, a logarithmic equation asks what power the base number must be raised to in order to get a certain result.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the exponent. You can do this by using the properties of logarithms, such as:

• log_a b = x => a^x = b
• log_a b = log_a c => b = c
• log_a b = log_c d => log_a b = log_c d

Q: What is the base of a logarithmic equation?

A: The base of a logarithmic equation is the number that the logarithm is being taken of. For example, in the equation log_2 4 = 2, the base is 2.

Q: What is the result of a logarithmic equation?

A: The result of a logarithmic equation is the number that the logarithm is being taken of. For example, in the equation log_2 4 = 2, the result is 4.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the exponent. You can do this by using a calculator or by using the properties of logarithms.

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of e (Euler's number).

Q: How do I convert a common logarithm to a natural logarithm?

A: To convert a common logarithm to a natural logarithm, you can use the following formula:

log_b x = ln(x) / ln(b)

Q: How do I convert a natural logarithm to a common logarithm?

A: To convert a natural logarithm to a common logarithm, you can use the following formula:

ln(x) = log_b x * ln(b)

Q: What are some common logarithmic equations?

A: Some common logarithmic equations include:

• log_a b = x => a^x = b
• log_a b = log_a c => b = c
• log_a b = log_c d => log_a b = log_c d

Q: How do I use logarithmic equations in real-life situations?

A: Logarithmic equations are used in many real-life situations, such as:

• Calculating the pH of a solution
• Determining the amount of time it takes for a population to grow or decline
• Calculating the interest rate on a loan
• Determining the amount of time it takes for a chemical reaction to occur

Conclusion

In conclusion, logarithmic equations are an important topic in mathematics that can be used to solve a wide range of problems. By understanding the properties of logarithmic equations and how to solve them, you can apply logarithmic equations to real-life situations.

Final Tips

• Make sure to understand the properties of logarithmic equations before trying to solve them.
• Use a calculator or a logarithmic table to evaluate logarithmic expressions.
• Practice solving logarithmic equations to become more comfortable with the concept.
• Apply logarithmic equations to real-life situations to see how they can be used to solve problems.