Which Exponential Equation Is Equivalent To The Logarithmic Equation Below? { \log 784 = A$}$A. ${ 784^{10} = A\$} B. { A^{10} = 784$}$C. ${ 10^a = 784\$} D. ${ 784^a = 10\$}

Mar 10, 2025 by ADMIN 175 views

Which Exponential Equation is Equivalent to the Logarithmic Equation Below?

Understanding Logarithmic and Exponential Equations

In mathematics, logarithmic and exponential equations are two fundamental concepts that are closely related. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. On the other hand, an exponential equation is an equation that involves an exponent, which is a power to which a base number is raised. In this article, we will explore which exponential equation is equivalent to the logarithmic equation $\log 784 = a$ .

The Relationship Between Logarithmic and Exponential Equations

To understand which exponential equation is equivalent to the logarithmic equation, we need to recall the relationship between logarithmic and exponential equations. The logarithmic equation $\log_b x = y$ is equivalent to the exponential equation $b^y = x$ . This means that if we have a logarithmic equation, we can rewrite it as an exponential equation by using the base and the exponent.

Rewriting the Logarithmic Equation as an Exponential Equation

Let's rewrite the logarithmic equation $\log 784 = a$ as an exponential equation. We can do this by using the base and the exponent. Since the logarithmic equation is in base 10, we can rewrite it as $10^a = 784$ . This means that the exponential equation $10^a = 784$ is equivalent to the logarithmic equation $\log 784 = a$ .

Evaluating the Answer Choices

Now that we have rewritten the logarithmic equation as an exponential equation, let's evaluate the answer choices.

A. $784^{10} = a$ : This is not equivalent to the logarithmic equation $\log 784 = a$ .
B. $a^{10} = 784$ : This is not equivalent to the logarithmic equation $\log 784 = a$ .
C. $10^a = 784$ : This is equivalent to the logarithmic equation $\log 784 = a$ .
D. $784^a = 10$ : This is not equivalent to the logarithmic equation $\log 784 = a$ .

Conclusion

In conclusion, the exponential equation that is equivalent to the logarithmic equation $\log 784 = a$ is $10^a = 784$ . This means that if we have a logarithmic equation, we can rewrite it as an exponential equation by using the base and the exponent.

Understanding the Properties of Logarithmic and Exponential Equations

Logarithmic and exponential equations have several properties that are important to understand. One of the most important properties is the relationship between logarithmic and exponential equations. This relationship is given by the equation $\log_b x = y \iff b^y = x$ . This means that if we have a logarithmic equation, we can rewrite it as an exponential equation by using the base and the exponent.

The Base of a Logarithmic Equation

The base of a logarithmic equation is an important concept to understand. The base of a logarithmic equation is the number that is used to raise the exponent. For example, in the logarithmic equation $\log 784 = a$ , the base is 10. This means that the exponent is 10 and the base is 10.

The Exponent of a Logarithmic Equation

The exponent of a logarithmic equation is also an important concept to understand. The exponent of a logarithmic equation is the power to which the base is raised. For example, in the logarithmic equation $\log 784 = a$ , the exponent is $a$ and the base is 10.

Solving Logarithmic and Exponential Equations

Solving logarithmic and exponential equations involves using the properties of logarithmic and exponential equations. One of the most important properties is the relationship between logarithmic and exponential equations. This relationship is given by the equation $\log_b x = y \iff b^y = x$ . This means that if we have a logarithmic equation, we can rewrite it as an exponential equation by using the base and the exponent.

Using the Change of Base Formula

The change of base formula is a useful formula for solving logarithmic and exponential equations. The change of base formula is given by the equation $\log_b x = \frac{\log_c x}{\log_c b}$ . This means that if we have a logarithmic equation with a base that is not 10, we can rewrite it as a logarithmic equation with a base of 10 by using the change of base formula.

Conclusion

In conclusion, logarithmic and exponential equations are two fundamental concepts in mathematics that are closely related. The relationship between logarithmic and exponential equations is given by the equation $\log_b x = y \iff b^y = x$ . This means that if we have a logarithmic equation, we can rewrite it as an exponential equation by using the base and the exponent. The base and exponent of a logarithmic equation are important concepts to understand, and solving logarithmic and exponential equations involves using the properties of logarithmic and exponential equations.

Final Answer

The final answer is C. $10^a = 784$ .
Q&A: Logarithmic and Exponential Equations

Understanding Logarithmic and Exponential Equations

In our previous article, we explored the relationship between logarithmic and exponential equations. We learned that a logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. On the other hand, an exponential equation is an equation that involves an exponent, which is a power to which a base number is raised. In this article, we will answer some frequently asked questions about logarithmic and exponential equations.

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 logarithmic equation $\log 784 = a$ is equivalent to the exponential equation $10^a = 784$ .

Q: How do I rewrite a logarithmic equation as an exponential equation?

A: To rewrite a logarithmic equation as an exponential equation, you need to use the base and the exponent. For example, the logarithmic equation $\log 784 = a$ can be rewritten as the exponential equation $10^a = 784$ .

Q: What is the base of a logarithmic equation?

A: The base of a logarithmic equation is the number that is used to raise the exponent. For example, in the logarithmic equation $\log 784 = a$ , the base is 10.

Q: What is the exponent of a logarithmic equation?

A: The exponent of a logarithmic equation is the power to which the base is raised. For example, in the logarithmic equation $\log 784 = a$ , the exponent is $a$ and the base is 10.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the properties of logarithmic equations. One of the most important properties is the relationship between logarithmic and exponential equations. This relationship is given by the equation $\log_b x = y \iff b^y = x$ . This means that if you have a logarithmic equation, you can rewrite it as an exponential equation by using the base and the exponent.

Q: What is the change of base formula?

A: The change of base formula is a useful formula for solving logarithmic and exponential equations. The change of base formula is given by the equation $\log_b x = \frac{\log_c x}{\log_c b}$ . This means that if you have a logarithmic equation with a base that is not 10, you can rewrite it as a logarithmic equation with a base of 10 by using the change of base formula.

Q: How do I use the change of base formula?

A: To use the change of base formula, you need to identify the base of the logarithmic equation and the base that you want to use. For example, if you have a logarithmic equation with a base of 2 and you want to use a base of 10, you can use the change of base formula to rewrite the equation.

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

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

Not using the correct base
Not using the correct exponent
Not using the change of base formula when necessary
Not checking the domain and range of the logarithmic equation

Conclusion

In conclusion, logarithmic and exponential equations are two fundamental concepts in mathematics that are closely related. The relationship between logarithmic and exponential equations is given by the equation $\log_b x = y \iff b^y = x$ . This means that if you have a logarithmic equation, you can rewrite it as an exponential equation by using the base and the exponent. By understanding the properties of logarithmic and exponential equations, you can solve a wide range of problems involving these equations.

Final Answer

The final answer is that logarithmic and exponential equations are two fundamental concepts in mathematics that are closely related. The relationship between logarithmic and exponential equations is given by the equation $\log_b x = y \iff b^y = x$ . This means that if you have a logarithmic equation, you can rewrite it as an exponential equation by using the base and the exponent.