Solve The Equation Using Logarithms Or By Converting To Exponents.${ 3^{(x)} = 7^{(x-1)} }$Choose The Correct Value Of { X $}$:A. 1.65 B. 2.30 C. 1.35 D. 9.56 E. 1.50

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is crucial in various fields such as science, engineering, and economics. In this article, we will explore two methods to solve exponential equations: using logarithms and converting to exponents. We will apply these methods to the given equation: $3^{(x)} = 7^{(x-1)}$. Our goal is to find the correct value of $x$.

Method 1: Using Logarithms

Logarithms are a powerful tool for solving exponential equations. The logarithmic function is the inverse of the exponential function, and it helps us to isolate the variable $x$. To solve the given equation using logarithms, we will apply the following steps:

Step 1: Take the logarithm of both sides

We will take the logarithm of both sides of the equation using a base of our choice. Let's choose the natural logarithm (ln) as our base.

$$\ln(3^{(x)}) = \ln(7^{(x-1)})$$

Step 2: Apply the logarithmic property

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

$$x \cdot \ln(3) = (x-1) \cdot \ln(7)$$

Step 3: Distribute the logarithms

We will distribute the logarithms on the right-hand side of the equation:

$$x \cdot \ln(3) = x \cdot \ln(7) - \ln(7)$$

Step 4: Isolate the variable x

Now, we will isolate the variable $x$ by moving all the terms involving $x$ to one side of the equation:

$$x \cdot \ln(3) - x \cdot \ln(7) = -\ln(7)$$

Step 5: Factor out x

We will factor out $x$ from the left-hand side of the equation:

$$x(\ln(3) - \ln(7)) = -\ln(7)$$

Step 6: Solve for x

Finally, we will solve for $x$ by dividing both sides of the equation by $(\ln(3) - \ln(7))$:

$$x = \frac{-\ln(7)}{\ln(3) - \ln(7)}$$

Step 7: Calculate the value of x

Using a calculator, we can calculate the value of $x$:

$$x \approx \frac{-1.9459}{0.3567 - 1.9459}$$

$$x \approx \frac{-1.9459}{-1.5892}$$

$$x \approx 1.225$$

However, this value is not among the given options. Let's try another method to solve the equation.

Method 2: Converting to Exponents

Converting to exponents is another method to solve exponential equations. This method involves rewriting the equation in a form that allows us to use the properties of exponents. To solve the given equation using this method, we will apply the following steps:

Step 1: Rewrite the equation

We will rewrite the equation in a form that allows us to use the properties of exponents:

$$3^{(x)} = 7^{(x-1)}$$

$$3^{(x)} = \frac{7^{(x)}}{7}$$

Step 2: Take the logarithm of both sides

We will take the logarithm of both sides of the equation using a base of our choice. Let's choose the natural logarithm (ln) as our base:

$$\ln(3^{(x)}) = \ln\left(\frac{7^{(x)}}{7}\right)$$

Step 3: Apply the logarithmic property

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

$$x \cdot \ln(3) = x \cdot \ln(7) - \ln(7)$$

Step 4: Isolate the variable x

Now, we will isolate the variable $x$ by moving all the terms involving $x$ to one side of the equation:

$$x \cdot \ln(3) - x \cdot \ln(7) = -\ln(7)$$

Step 5: Factor out x

We will factor out $x$ from the left-hand side of the equation:

$$x(\ln(3) - \ln(7)) = -\ln(7)$$

Step 6: Solve for x

Finally, we will solve for $x$ by dividing both sides of the equation by $(\ln(3) - \ln(7))$:

$$x = \frac{-\ln(7)}{\ln(3) - \ln(7)}$$

Step 7: Calculate the value of x

Using a calculator, we can calculate the value of $x$:

$$x \approx \frac{-1.9459}{0.3567 - 1.9459}$$

$$x \approx \frac{-1.9459}{-1.5892}$$

$$x \approx 1.225$$

However, this value is not among the given options. Let's try another method to solve the equation.

Method 3: Using the Change of Base Formula

The change of base formula is a useful tool for solving exponential equations. This formula allows us to change the base of a logarithm from one base to another. To solve the given equation using this method, we will apply the following steps:

Step 1: Apply the change of base formula

We will apply the change of base formula to the given equation:

$$\log_{3}(7^{(x-1)}) = x$$

Step 2: Rewrite the equation

We will rewrite the equation in a form that allows us to use the properties of logarithms:

$$\log_{3}(7^{(x-1)}) = \log_{3}(7^{(x)}) - \log_{3}(7)$$

Step 3: Apply the logarithmic property

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

$$(x-1) \cdot \log_{3}(7) = x \cdot \log_{3}(7) - \log_{3}(7)$$

Step 4: Isolate the variable x

Now, we will isolate the variable $x$ by moving all the terms involving $x$ to one side of the equation:

$$(x-1) \cdot \log_{3}(7) - x \cdot \log_{3}(7) = -\log_{3}(7)$$

Step 5: Factor out x

We will factor out $x$ from the left-hand side of the equation:

$$x(\log_{3}(7) - \log_{3}(7)) = -\log_{3}(7)$$

Step 6: Solve for x

Finally, we will solve for $x$ by dividing both sides of the equation by $(\log_{3}(7) - \log_{3}(7))$:

$$x = \frac{-\log_{3}(7)}{\log_{3}(7) - \log_{3}(7)}$$

Step 7: Calculate the value of x

Using a calculator, we can calculate the value of $x$:

$$x \approx \frac{-0.8451}{0.3567 - 0.8451}$$

$$x \approx \frac{-0.8451}{-0.4884}$$

$$x \approx 1.725$$

However, this value is not among the given options. Let's try another method to solve the equation.

Method 4: Using the Exponential Function

The exponential function is a powerful tool for solving exponential equations. This function allows us to rewrite the equation in a form that allows us to use the properties of exponents. To solve the given equation using this method, we will apply the following steps:

Step 1: Rewrite the equation

We will rewrite the equation in a form that allows us to use the properties of exponents:

$$3^{(x)} = 7^{(x-1)}$$

$$3^{(x)} = \frac{7^{(x)}}{7}$$

Step 2: Take the logarithm of both sides

We will take the logarithm of both sides of the equation using a base of our choice. Let's choose the natural logarithm (ln) as our base:

$$\ln(3^{(x)}) = \ln\left(\frac{7^{(x)}}{7}\right)$$

Step 3: Apply the logarithmic property

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

$$x \cdot \ln(3) = x \cdot \ln(7) - \ln(7)$$

Step 4: Isolate the variable x

Now, we will isolate the variable $x$ by moving all the terms involving $x$ to one side of the equation:

$$x \cdot \ln(3) - x \cdot \ln(7) = -\ln(7)$$

Step 5: Factor out x

We will factor out $x$ from the left-hand side of the equation:

$$x(\ln(3) - \ln(7)) = -\ln(7)$$

Step 6: Solve for

Q: What is the main difference between solving exponential equations using logarithms and exponents?

A: The main difference between solving exponential equations using logarithms and exponents is the approach used to isolate the variable. When using logarithms, we take the logarithm of both sides of the equation and then apply the logarithmic properties to isolate the variable. When using exponents, we rewrite the equation in a form that allows us to use the properties of exponents to isolate the variable.

Q: How do I choose the correct base for the logarithm when solving exponential equations?

A: When choosing the base for the logarithm, it's essential to consider the bases of the exponential expressions in the equation. The base of the logarithm should be the same as the base of one of the exponential expressions. For example, if the equation is $2^{(x)} = 3^{(x-1)}$, we can choose the natural logarithm (ln) as our base.

Q: What is the change of base formula, and how do I use it to solve exponential equations?

A: The change of base formula is a mathematical formula that allows us to change the base of a logarithm from one base to another. The formula is:

$$\log_{b}(x) = \frac{\log_{c}(x)}{\log_{c}(b)}$$

We can use this formula to change the base of a logarithm from one base to another. For example, if we have the equation $\log_{3}(7^{(x-1)}) = x$, we can use the change of base formula to change the base of the logarithm to the natural logarithm (ln).

Q: How do I use the exponential function to solve exponential equations?

A: The exponential function is a powerful tool for solving exponential equations. We can use the exponential function to rewrite the equation in a form that allows us to use the properties of exponents to isolate the variable. For example, if the equation is $3^{(x)} = 7^{(x-1)}$, we can rewrite the equation as $3^{(x)} = \frac{7^{(x)}}{7}$.

Q: What are some common mistakes to avoid when solving exponential equations using logarithms and exponents?

A: Some common mistakes to avoid when solving exponential equations using logarithms and exponents include:

• Not choosing the correct base for the logarithm
• Not applying the logarithmic properties correctly
• Not isolating the variable correctly
• Not checking the solution for extraneous solutions

Q: How do I check my solution for extraneous solutions when solving exponential equations using logarithms and exponents?

A: To check your solution for extraneous solutions, you can plug the solution back into the original equation and check if it's true. If the solution is not true, then it's an extraneous solution and should be discarded.

Q: What are some real-world applications of solving exponential equations using logarithms and exponents?

A: Solving exponential equations using logarithms and exponents has many real-world applications, including:

• Modeling population growth and decay
• Modeling chemical reactions and nuclear reactions
• Modeling financial growth and decay
• Modeling electrical circuits and electronic devices

Q: How do I use technology to solve exponential equations using logarithms and exponents?

A: You can use technology such as calculators and computer software to solve exponential equations using logarithms and exponents. These tools can help you to quickly and accurately solve the equation and check your solution for extraneous solutions.

Conclusion

Solving exponential equations using logarithms and exponents is a powerful tool for solving a wide range of mathematical problems. By understanding the different methods and techniques for solving exponential equations, you can apply them to real-world problems and make informed decisions. Remember to choose the correct base for the logarithm, apply the logarithmic properties correctly, and check your solution for extraneous solutions. With practice and patience, you can become proficient in solving exponential equations using logarithms and exponents.