Select The Correct Answer.Which Exponential Equation Is Equivalent To This Logarithmic Equation?$\log _5 X-\log _5 25=7$A. $7^9=x$ B. $5^9=x$ C. $5^5=x$ D. $7^5=x$

Introduction

Logarithmic and exponential equations are fundamental concepts in mathematics, and understanding how to solve them is crucial for success in various fields, including science, engineering, and finance. In this article, we will focus on solving exponential equations that are equivalent to logarithmic equations. We will use the given logarithmic equation $\log _5 x-\log _5 25=7$ as an example and explore the different options to find the correct answer.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The logarithmic equation $\log _5 x-\log _5 25=7$ can be rewritten using the properties of logarithms. Specifically, we can use the property that $\log _a b - \log _a c = \log _a \frac{b}{c}$ to simplify the equation.

Simplifying the Logarithmic Equation

Using the property mentioned above, we can rewrite the logarithmic equation as:

$$\log _5 x - \log _5 25 = \log _5 \frac{x}{25} = 7$$

This simplifies to:

$$\log _5 \frac{x}{25} = 7$$

Converting to Exponential Form

To convert the logarithmic equation to exponential form, we can use the definition of a logarithm, which states that $\log _a b = c$ is equivalent to $a^c = b$. Applying this definition to our equation, we get:

$$5^7 = \frac{x}{25}$$

Solving for x

To solve for x, we can multiply both sides of the equation by 25:

$$5^7 \cdot 25 = x$$

Using the property that $a^m \cdot a^n = a^{m+n}$, we can simplify the left-hand side of the equation:

$$5^7 \cdot 5^2 = x$$

This simplifies to:

$$5^{7+2} = x$$

$$5^9 = x$$

Conclusion

In conclusion, we have shown that the exponential equation $5^9 = x$ is equivalent to the logarithmic equation $\log _5 x-\log _5 25=7$. This demonstrates the importance of understanding the properties of logarithms and how to convert between logarithmic and exponential equations.

Answer

The correct answer is:

• B. $5^9=x$

This is the only option that matches the solution we obtained by converting the logarithmic equation to exponential form.

Additional Tips and Tricks

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

• Use the properties of logarithms to simplify equations and make them easier to solve.
• Convert between logarithmic and exponential form using the definition of a logarithm.
• Be careful when multiplying or dividing both sides of an equation by a value that contains a variable.
• Use the properties of exponents to simplify expressions and make them easier to work with.

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

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. An exponential equation, on the other hand, is an equation that involves an exponent. For example, the equation $\log _5 x = 7$ is a logarithmic equation, while the equation $5^x = 7$ is an exponential equation.

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

A: To convert a logarithmic equation to exponential form, you can use the definition of a logarithm, which states that $\log _a b = c$ is equivalent to $a^c = b$. For example, the equation $\log _5 x = 7$ can be rewritten as $5^7 = x$.

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

A: To convert an exponential equation to logarithmic form, you can use the definition of a logarithm, which states that $a^c = b$ is equivalent to $\log _a b = c$. For example, the equation $5^x = 7$ can be rewritten as $\log _5 7 = x$.

Q: What are some common properties of logarithms that I should know?

A: Some common properties of logarithms include:

• $\log _a b - \log _a c = \log _a \frac{b}{c}$
• $\log _a b + \log _a c = \log _a (b \cdot c)$
• $\log _a b^c = c \log _a b$
• $\log _a \frac{1}{b} = -\log _a b$

Q: How do I simplify a logarithmic equation using the properties of logarithms?

A: To simplify a logarithmic equation using the properties of logarithms, you can use the properties mentioned above to combine or eliminate logarithms. For example, the equation $\log _5 x - \log _5 25 = 7$ can be simplified using the property $\log _a b - \log _a c = \log _a \frac{b}{c}$.

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:

• Forgetting to use the definition of a logarithm to convert between logarithmic and exponential form.
• Not using the properties of logarithms to simplify equations.
• Making errors when multiplying or dividing both sides of an equation by a value that contains a variable.
• Not checking the domain of a logarithmic function.

Q: How do I check the domain of a logarithmic function?

A: To check the domain of a logarithmic function, you need to ensure that the argument of the logarithm is positive. For example, the function $\log _5 x$ is defined only for $x > 0$.

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

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

• Finance: Logarithmic and exponential equations are used to calculate interest rates, investment returns, and other financial metrics.
• Science: Logarithmic and exponential equations are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Logarithmic and exponential equations are used to design and optimize systems, such as electronic circuits and mechanical systems.

By understanding logarithmic and exponential equations, you can develop a deeper appreciation for the mathematical concepts that underlie many real-world applications.