Solve For $x$: $9^x = 4^{(x+3)}$Round Your Answer To The Nearest Thousandth. Type Your Numerical Answer Below.$\square$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and their properties. In this article, we will focus on solving the equation $9^x = 4^{(x+3)}$ and provide a step-by-step guide on how to approach such problems.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be challenging to solve. However, with the right approach and techniques, we can simplify these equations and find the value of the variable. In this case, we have the equation $9^x = 4^{(x+3)}$, where $x$ is the variable we need to solve for.

Taking the Logarithm of Both Sides

One of the most effective ways to solve exponential equations is to take the logarithm of both sides. This allows us to use the properties of logarithms to simplify the equation and isolate the variable. In this case, we can take the logarithm of both sides of the equation $9^x = 4^{(x+3)}$.

$$\log(9^x) = \log(4^{(x+3)})$$

Using the property of logarithms that states $\log(a^b) = b\log(a)$, we can simplify the equation as follows:

$$x\log(9) = (x+3)\log(4)$$

Simplifying the Equation

Now that we have simplified the equation, we can start to isolate the variable $x$. To do this, we can expand the right-hand side of the equation and then combine like terms.

$$x\log(9) = x\log(4) + 3\log(4)$$

Subtracting $x\log(4)$ from both sides of the equation gives us:

$$x\log(9) - x\log(4) = 3\log(4)$$

Factoring out $x$ from the left-hand side of the equation gives us:

$$x(\log(9) - \log(4)) = 3\log(4)$$

Dividing Both Sides by the Coefficient

Now that we have isolated the variable $x$, we can divide both sides of the equation by the coefficient of $x$ to solve for $x$.

$$x = \frac{3\log(4)}{\log(9) - \log(4)}$$

Using a Calculator to Find the Value of $x$

To find the value of $x$, we can use a calculator to evaluate the expression on the right-hand side of the equation.

$$x = \frac{3\log(4)}{\log(9) - \log(4)} \approx \frac{3(0.60206)}{0.95424 - 0.60206} \approx \frac{1.80618}{0.35218} \approx 5.119$$

Conclusion

In this article, we have shown how to solve the exponential equation $9^x = 4^{(x+3)}$ using the properties of logarithms. By taking the logarithm of both sides of the equation and simplifying it, we were able to isolate the variable $x$ and find its value. We also used a calculator to evaluate the expression on the right-hand side of the equation and found that $x \approx 5.119$.

Tips and Tricks

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

• Take the logarithm of both sides of the equation to simplify it and isolate the variable.
• Use the properties of logarithms to simplify the equation and combine like terms.
• Factor out the coefficient of the variable to isolate it.
• Use a calculator to evaluate the expression on the right-hand side of the equation.

By following these tips and tricks, you can solve exponential equations with ease and confidence.

Common Mistakes to Avoid

When solving exponential equations, it's essential to avoid the following common mistakes:

• Not taking the logarithm of both sides of the equation.
• Not using the properties of logarithms to simplify the equation.
• Not factoring out the coefficient of the variable.
• Not using a calculator to evaluate the expression on the right-hand side of the equation.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

Exponential equations have numerous real-world applications in fields such as finance, economics, and science. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

In finance, exponential growth is used to calculate compound interest, while in economics, it's used to model economic growth and inflation. In science, exponential decay is used to model radioactive decay and chemical reactions.

Conclusion

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Introduction

In our previous article, we discussed how to solve exponential equations using the properties of logarithms. However, we understand that some readers may still have questions or need further clarification on certain concepts. In this article, we will address some of the most frequently asked questions about solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. For example, $9^x = 4^{(x+3)}$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to take the logarithm of both sides of the equation and then use the properties of logarithms to simplify it and isolate the variable.

Q: What is the logarithm of a number?

A: The logarithm of a number is the power to which a base number must be raised to produce that number. For example, $\log(100) = 2$ because $10^2 = 100$.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• $\log(a^b) = b\log(a)$
• $\log(a) + \log(b) = \log(ab)$
• $\log(a) - \log(b) = \log(\frac{a}{b})$

Q: How do I use the properties of logarithms to simplify an exponential equation?

A: To use the properties of logarithms to simplify an exponential equation, you need to take the logarithm of both sides of the equation and then apply the properties of logarithms to simplify it.

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

A: A logarithmic equation is an equation that involves a variable in the logarithm, while an exponential equation is an equation that involves a variable in the exponent.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to use the properties of logarithms to isolate the variable.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not taking the logarithm of both sides of the equation
• Not using the properties of logarithms to simplify the equation
• Not factoring out the coefficient of the variable
• Not using a calculator to evaluate the expression on the right-hand side of the equation

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

A: Exponential equations have numerous real-world applications in fields such as finance, economics, and science. For example, exponential growth and decay are used to model population growth, chemical reactions, and financial investments.

Q: How do I use a calculator to evaluate an expression on the right-hand side of an exponential equation?

A: To use a calculator to evaluate an expression on the right-hand side of an exponential equation, you need to follow these steps:

1. Enter the expression into the calculator.
2. Press the "calculate" button.
3. Read the result.

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

A: A linear equation is an equation that involves a variable in a linear relationship, while an exponential equation is an equation that involves a variable in an exponential relationship.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to use algebraic methods to isolate the variable.

Conclusion

In this article, we have addressed some of the most frequently asked questions about solving exponential equations. We hope that this article has provided you with a better understanding of exponential equations and how to solve them. If you have any further questions, please don't hesitate to ask.

Additional Resources

For further information on solving exponential equations, we recommend the following resources:

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations

Final Tips

• Practice solving exponential equations regularly to build your skills and confidence.
• Use a calculator to evaluate expressions on the right-hand side of exponential equations.
• Avoid common mistakes such as not taking the logarithm of both sides of the equation and not using the properties of logarithms to simplify the equation.

By following these tips and using the resources provided, you can become proficient in solving exponential equations and apply them to real-world problems.