Solve 3 X + 1 = 15 3^{x+1}=15 3 X + 1 = 15 For X X X Using The Change Of Base Formula: Log ⁡ B Y = Log ⁡ Y Log ⁡ B \log_b Y = \frac{\log Y}{\log B} Lo G B ​ Y = L O G B L O G Y ​ A. -0.594316 B. 1.405684 C. 1.469743 D. 3.469743

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is crucial in various fields such as physics, engineering, and computer science. In this article, we will focus on solving the exponential equation $3^{x+1}=15$ using the change of base formula. The change of base formula is a powerful tool that allows us to rewrite logarithmic expressions in a more convenient form.

The Change of Base Formula

The change of base formula is given by:

$\log_b y = \frac{\log y}{\log b}$

This formula allows us to rewrite a logarithmic expression in terms of a common logarithm (base 10) or a natural logarithm (base e). The change of base formula is a fundamental concept in mathematics and is widely used in various fields.

Solving the Exponential Equation

Now, let's focus on solving the exponential equation $3^{x+1}=15$. To solve this equation, we can use the change of base formula to rewrite the equation in a more convenient form.

First, we can rewrite the equation as:

$3^{x+1} = 15$

Next, we can take the logarithm of both sides of the equation using the change of base formula:

$\log_3 3^{x+1} = \log_3 15$

Using the property of logarithms that states $\log_b b^x = x$, we can simplify the left-hand side of the equation:

$x+1 = \log_3 15$

Now, we can use the change of base formula to rewrite the right-hand side of the equation in terms of a common logarithm (base 10):

$x+1 = \frac{\log 15}{\log 3}$

Simplifying the Equation

To simplify the equation, we can use the property of logarithms that states $\log_b a = \frac{\log a}{\log b}$. This allows us to rewrite the equation as:

$x+1 = \frac{\log 15}{\log 3}$

$x+1 = \frac{\log 15}{\log 3}$

$x = \frac{\log 15}{\log 3} - 1$

Evaluating the Expression

To evaluate the expression, we can use a calculator to find the values of the logarithms:

$\log 15 \approx 1.1761$

$\log 3 \approx 0.4771$

Now, we can substitute these values into the expression:

$x \approx \frac{1.1761}{0.4771} - 1$

$x \approx 2.4673 - 1$

$x \approx 1.4673$

Conclusion

In this article, we used the change of base formula to solve the exponential equation $3^{x+1}=15$. We first rewrote the equation in a more convenient form using the change of base formula, and then simplified the equation using the properties of logarithms. Finally, we evaluated the expression using a calculator to find the value of x.

Answer

The final answer is:

$\boxed{1.469743}$

$$

Introduction

In our previous article, we used the change of base formula to solve the exponential equation $3^{x+1}=15$. In this article, we will provide a Q&A section to help you better understand the concept and solve similar problems.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to rewrite a logarithmic expression in terms of a common logarithm (base 10) or a natural logarithm (base e). The formula is given by:

$\log_b y = \frac{\log y}{\log b}$

Q: How do I use the change of base formula to solve an exponential equation?

A: To use the change of base formula to solve an exponential equation, follow these steps:

1. Rewrite the equation in a more convenient form using the change of base formula.
2. Simplify the equation using the properties of logarithms.
3. Evaluate the expression using a calculator to find the value of x.

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

A: Here are some common logarithmic properties that you should know:

• $\log_b b^x = x$
• $\log_b a^x = x \log_b a$
• $\log_b (a \cdot b) = \log_b a + \log_b b$
• $\log_b (a / b) = \log_b a - \log_b b$

Q: How do I choose the base of the logarithm?

A: The base of the logarithm is usually chosen to be a convenient number that makes the calculation easier. For example, if the equation involves a base of 10, it's often easier to use the common logarithm (base 10). If the equation involves a base of e, it's often easier to use the natural logarithm (base e).

Q: Can I use the change of base formula to solve logarithmic equations?

A: Yes, you can use the change of base formula to solve logarithmic equations. The formula can be used to rewrite a logarithmic expression in terms of a common logarithm (base 10) or a natural logarithm (base e).

Q: What are some common mistakes to avoid when using the change of base formula?

A: Here are some common mistakes to avoid when using the change of base formula:

• Not rewriting the equation in a more convenient form using the change of base formula.
• Not simplifying the equation using the properties of logarithms.
• Not evaluating the expression using a calculator to find the value of x.

Q: Can I use the change of base formula to solve equations with negative exponents?

A: Yes, you can use the change of base formula to solve equations with negative exponents. The formula can be used to rewrite a logarithmic expression in terms of a common logarithm (base 10) or a natural logarithm (base e).

Conclusion

In this article, we provided a Q&A section to help you better understand the concept of using the change of base formula to solve exponential equations. We covered common logarithmic properties, choosing the base of the logarithm, and common mistakes to avoid. We hope this article has been helpful in your understanding of the change of base formula.

Additional Resources

If you want to learn more about the change of base formula and how to use it to solve exponential equations, here are some additional resources:

• Khan Academy: Change of Base Formula
• Mathway: Change of Base Formula
• Wolfram Alpha: Change of Base Formula

We hope this article has been helpful in your understanding of the change of base formula. If you have any further questions or need additional help, please don't hesitate to ask.