Which Logarithmic Equation Is Equivalent To The Exponential Equation Below?$e^{2x} = 7$A. $\ln 2x = 7$B. $\log 7 = 2x$C. $\ln 7 = 2x$D. $\log 2x = 7$

Introduction

Logarithmic equations and exponential equations are two fundamental concepts in mathematics that are closely related. In this article, we will explore how to convert an exponential equation into a logarithmic equation and vice versa. We will use the given exponential equation $e^{2x} = 7$ as an example to demonstrate the process.

Understanding Exponential and Logarithmic Equations

Exponential equations involve a base raised to a power, while logarithmic equations involve the inverse operation of exponentiation. The exponential equation $e^{2x} = 7$ can be rewritten as $e^{2x} - 7 = 0$. This equation can be solved using various methods, including the quadratic formula.

Converting Exponential to Logarithmic

To convert the exponential equation $e^{2x} = 7$ into a logarithmic equation, we need to use the definition of logarithms. The logarithmic equation is defined as the inverse operation of exponentiation. In other words, if $a^x = b$, then $\log_a b = x$. We can rewrite the exponential equation as $\log_e (e^{2x}) = \log_e 7$. Using the property of logarithms that $\log_a a^x = x$, we can simplify the equation to $2x = \log_e 7$.

Simplifying the Logarithmic Equation

We can simplify the logarithmic equation $2x = \log_e 7$ by using the property of logarithms that $\log_a b = \frac{\log_c b}{\log_c a}$. In this case, we can rewrite the equation as $2x = \frac{\log 7}{\log e}$. Since $\log e = 1$, we can simplify the equation to $2x = \log 7$.

Evaluating the Options

Now that we have simplified the logarithmic equation, we can evaluate the options given in the problem. The options are:

A. $\ln 2x = 7$ B. $\log 7 = 2x$ C. $\ln 7 = 2x$ D. $\log 2x = 7$

We can see that option C is the only option that matches the simplified logarithmic equation $2x = \log 7$.

Conclusion

In conclusion, we have demonstrated how to convert an exponential equation into a logarithmic equation and vice versa. We used the given exponential equation $e^{2x} = 7$ as an example to demonstrate the process. We simplified the logarithmic equation using the properties of logarithms and evaluated the options given in the problem. The correct answer is option C, $\ln 7 = 2x$.

Final Answer

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

Additional Tips and Tricks

• When converting an exponential equation into a logarithmic equation, use the definition of logarithms to rewrite the equation.
• Use the properties of logarithms to simplify the equation.
• Evaluate the options given in the problem to determine the correct answer.

Common Mistakes to Avoid

• Do not confuse the base of the logarithm with the exponent.
• Do not forget to use the properties of logarithms to simplify the equation.
• Do not evaluate the options without simplifying the logarithmic equation.

Real-World Applications

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 chemical reactions.
• Engineering: Logarithmic equations are used to design electronic circuits and optimize system performance.

Conclusion

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

A: An exponential equation involves a base raised to a power, while a logarithmic equation involves the inverse operation of exponentiation. In other words, if $a^x = b$, then $\log_a b = x$.

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

A: To convert an exponential equation into a logarithmic equation, use the definition of logarithms to rewrite the equation. For example, if $e^{2x} = 7$, we can rewrite it as $\log_e (e^{2x}) = \log_e 7$.

Q: What are the properties of logarithms that I need to know?

A: There are several properties of logarithms that you need to know:

• $\log_a a^x = x$
• $\log_a b = \frac{\log_c b}{\log_c a}$
• $\log_a b = \frac{1}{\log_b a}$

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, use the properties of logarithms to rewrite the equation. For example, if $2x = \log 7$, we can rewrite it as $2x = \frac{\log 7}{\log e}$.

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

A: A natural logarithm is a logarithm with a base of $e$, while a common logarithm is a logarithm with a base of 10.

Q: How do I evaluate a logarithmic equation?

A: To evaluate a logarithmic equation, use the properties of logarithms to rewrite the equation. For example, if $\log 7 = 2x$, we can rewrite it as $7 = e^{2x}$.

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:

• Confusing the base of the logarithm with the exponent
• Forgetting to use the properties of logarithms to simplify the equation
• Evaluating the options without simplifying the logarithmic equation

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 chemical reactions.
• Engineering: Logarithmic equations are used to design electronic circuits and optimize system performance.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, use a graphing calculator or software to plot the function. You can also use the properties of logarithms to rewrite the function in a more convenient form.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

• $\log x$
• $\ln x$
• $\log_b x$

Q: How do I solve a logarithmic equation with a base other than $e$ or 10?

A: To solve a logarithmic equation with a base other than $e$ or 10, use the properties of logarithms to rewrite the equation. For example, if $\log_3 7 = 2x$, we can rewrite it as $7 = 3^{2x}$.

Q: What are some tips for solving logarithmic equations?

A: Some tips for solving logarithmic equations include:

• Use the properties of logarithms to rewrite the equation
• Simplify the equation using the properties of logarithms
• Evaluate the options without simplifying the logarithmic equation

Q: How do I check my answer to a logarithmic equation?

A: To check your answer to a logarithmic equation, use the properties of logarithms to rewrite the equation. For example, if $\log 7 = 2x$, we can rewrite it as $7 = e^{2x}$. If the equation is true, then your answer is correct.