Select The Correct Answer.Which Logarithmic Equation Is Equivalent To This Exponential Equation? 2 , 400 = 7 , 500 ( 10 ) − X 2,400 = 7,500(10)^{-x} 2 , 400 = 7 , 500 ( 10 ) − X A. X = \log \left(-\frac{25}{8}\right ]B. X = -\log \left(\frac{25}{8}\right ]C. $x = \log

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will explore how to solve logarithmic equations, with a focus on identifying the correct equivalent form of a given exponential equation.

Understanding Logarithmic Equations

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

$$\log_b(x) = y \iff b^y = x$$

where $b$ is the base of the logarithm, and $x$ is the input value.

Converting Exponential Equations to Logarithmic Equations

To convert an exponential equation to a logarithmic equation, we can use the following property:

$$b^x = y \iff x = \log_b(y)$$

This property allows us to rewrite an exponential equation in logarithmic form.

Solving the Given Exponential Equation

The given exponential equation is:

$$2,400 = 7,500(10)^{-x}$$

To solve this equation, we can start by isolating the exponential term:

$$\frac{2,400}{7,500} = (10)^{-x}$$

Simplifying the left-hand side, we get:

$$\frac{1}{3} = (10)^{-x}$$

Now, we can take the logarithm of both sides to get:

$$\log\left(\frac{1}{3}\right) = -x \log(10)$$

Using the property of logarithms that $\log_b(b) = 1$, we can rewrite the equation as:

$$\log\left(\frac{1}{3}\right) = -x$$

Evaluating the Logarithmic Expression

To evaluate the logarithmic expression, we can use the fact that $\log_b(b) = 1$:

$$\log\left(\frac{1}{3}\right) = \log(10^{-x})$$

Using the property of logarithms that $\log_b(b^x) = x$, we can rewrite the equation as:

$$-x = \log\left(\frac{1}{3}\right)$$

Comparing the Options

Now, let's compare the options:

A. $x = \log \left(-\frac{25}{8}\right)$

B. $x = -\log \left(\frac{25}{8}\right)$

C. $x = \log \left(\frac{1}{3}\right)$

The correct option is:

C. $x = \log \left(\frac{1}{3}\right)$

Conclusion

In this article, we have explored how to solve logarithmic equations by converting exponential equations to logarithmic form. We have also evaluated the logarithmic expression and compared the options to find the correct answer. By following these steps, you can develop a deep understanding of logarithmic equations and improve your problem-solving skills.

Additional Tips and Tricks

• When solving logarithmic equations, make sure to isolate the logarithmic term.
• Use the properties of logarithms to simplify the equation.
• Evaluate the logarithmic expression using the properties of logarithms.
• Compare the options carefully to find the correct answer.

Common Mistakes to Avoid

• Failing to isolate the logarithmic term.
• Not using the properties of logarithms to simplify the equation.
• Evaluating the logarithmic expression incorrectly.
• Not comparing the options carefully.

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 decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Final Thoughts

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

A: A logarithmic equation is an equation that involves a logarithmic function, which is the inverse of an exponential function. An exponential equation, on the other hand, is an equation that involves an exponential function.

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

A: To convert an exponential equation to a logarithmic equation, you can use the following property:

$$b^x = y \iff x = \log_b(y)$$

This property allows you to rewrite an exponential equation in logarithmic form.

Q: What is the base of a logarithmic equation?

A: The base of a logarithmic equation is the number that is raised to a power in the exponential form of the equation. For example, in the equation $\log_2(x) = y$, the base is 2.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the properties of logarithms, such as the product rule and the quotient rule. For example, to evaluate the expression $\log_2(3x)$, you can use the product rule to rewrite it as $\log_2(3) + \log_2(x)$.

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

A: A logarithmic function is a function that involves a logarithmic expression, while a logarithmic equation is an equation that involves a logarithmic expression. For example, the function $f(x) = \log_2(x)$ is a logarithmic function, while the equation $\log_2(x) = 3$ is a logarithmic equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the following steps:

1. Isolate the logarithmic term.
2. Use the properties of logarithms to simplify the equation.
3. Evaluate the logarithmic expression.
4. Solve for the variable.

Q: What is the logarithmic form of the equation $2^x = 8$?

A: The logarithmic form of the equation $2^x = 8$ is $x = \log_2(8)$.

Q: How do I use a calculator to evaluate a logarithmic expression?

A: To use a calculator to evaluate a logarithmic expression, you can enter the expression into the calculator and press the "log" button. For example, to evaluate the expression $\log_2(8)$, you can enter the expression into the calculator and press the "log" button.

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.

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 property:

$$\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$$

This property allows you to convert a common logarithm to a natural logarithm.

Q: What is the logarithmic form of the equation $e^x = 10$?

A: The logarithmic form of the equation $e^x = 10$ is $x = \ln(10)$.

Q: How do I use a calculator to evaluate a natural logarithmic expression?

A: To use a calculator to evaluate a natural logarithmic expression, you can enter the expression into the calculator and press the "ln" button. For example, to evaluate the expression $\ln(10)$, you can enter the expression into the calculator and press the "ln" button.

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

A: A logarithmic equation is an equation that involves a logarithmic expression, while a quadratic equation is an equation that involves a quadratic expression. For example, the equation $\log_2(x) = 3$ is a logarithmic equation, while the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Factor the quadratic expression.
2. Use the quadratic formula to find the solutions.
3. Simplify the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to find the solutions to a quadratic equation. The formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

This formula allows you to find the solutions to a quadratic equation.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you can plug in the values of $a$, $b$, and $c$ into the formula and simplify the expression. For example, to solve the equation $x^2 + 4x + 4 = 0$, you can plug in the values $a = 1$, $b = 4$, and $c = 4$ into the formula and simplify the expression.