Consider The Exponential Equation $4^x = 30$.(a) Between What Two Consecutive Integers Must The Solution To This Equation Lie? Explain Your Reasoning.(b) Write $\log(4^x)$ As An Equivalent Product Using The Third Logarithm Law.

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic functions and properties. In this article, we will explore the solution to the exponential equation $4^x = 30$ and provide a step-by-step guide on how to solve it.

Part (a): Finding the Range of the Solution

To find the range of the solution, we need to determine between which two consecutive integers the solution must lie. We can start by rewriting the equation as $4^x = 30$.

Since $4^x$ is an exponential function, we know that it is increasing for all values of $x$. This means that as $x$ increases, $4^x$ also increases. Therefore, we can conclude that $x$ must be greater than the logarithm of 30 to the base 4.

We can calculate the logarithm of 30 to the base 4 using a calculator or by using the change of base formula:

$$\log_4(30) = \frac{\log(30)}{\log(4)}$$

Using a calculator, we get:

$$\log_4(30) \approx 2.95$$

Since $x$ must be greater than 2.95, we can conclude that the solution must lie between 2 and 3.

Part (b): Writing $\log(4^x)$ as an Equivalent Product

To write $\log(4^x)$ as an equivalent product using the third logarithm law, we need to recall the property of logarithms that states:

$$\log(a^b) = b \log(a)$$

Using this property, we can rewrite $\log(4^x)$ as:

$$\log(4^x) = x \log(4)$$

This is the equivalent product form of $\log(4^x)$.

The Third Logarithm Law

The third logarithm law states that:

$$\log(a) + \log(b) = \log(ab)$$

Using this law, we can rewrite $\log(4^x)$ as:

$$\log(4^x) = x \log(4) = \log(4^x)$$

This is the equivalent product form of $\log(4^x)$.

Conclusion

In this article, we have explored the solution to the exponential equation $4^x = 30$ and provided a step-by-step guide on how to solve it. We have also used the third logarithm law to rewrite $\log(4^x)$ as an equivalent product.

Key Takeaways

• The solution to the exponential equation $4^x = 30$ must lie between 2 and 3.
• The equivalent product form of $\log(4^x)$ is $x \log(4)$.
• The third logarithm law states that $\log(a) + \log(b) = \log(ab)$.

Further Reading

For further reading on exponential equations and logarithmic functions, we recommend the following resources:

• Exponential Equations
• Logarithmic Functions
• The Third Logarithm Law

Glossary

• Exponential Equation: An equation in which the variable appears as an exponent.
• Logarithmic Function: A function that is the inverse of an exponential function.
• Third Logarithm Law: A property of logarithms that states $\log(a) + \log(b) = \log(ab)$.

References

• [1] Exponential Equations
• [2] Logarithmic Functions
• [3] The Third Logarithm Law

About the Author

Introduction

In our previous article, we explored the solution to the exponential equation $4^x = 30$ and provided a step-by-step guide on how to solve it. We also used the third logarithm law to rewrite $\log(4^x)$ as an equivalent product. In this article, we will answer some frequently asked questions about exponential equations and logarithmic functions.

Q: What is an exponential equation?

A: An exponential equation is an equation in which the variable appears as an exponent. For example, $4^x = 30$ is an exponential equation because the variable $x$ appears as an exponent.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable by using logarithmic functions. You can use the property of logarithms that states $\log(a^b) = b \log(a)$ to rewrite the equation in a more manageable form.

Q: What is the third logarithm law?

A: The third logarithm law states that $\log(a) + \log(b) = \log(ab)$. This law allows you to combine two logarithmic expressions into a single logarithmic expression.

Q: How do I use the third logarithm law?

A: To use the third logarithm law, you need to identify the two logarithmic expressions that you want to combine. Then, you can use the law to rewrite the expressions as a single logarithmic expression.

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

A: A logarithmic function is the inverse of an exponential function. While an exponential function grows rapidly as the input increases, a logarithmic function grows slowly as the input increases.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to use a graphing calculator or a computer program. You can also use a table of values to create a graph.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

• $\log(x)$
• $\log_2(x)$
• $\log_3(x)$
• $\log_4(x)$

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to use the definition of a logarithm. For example, $\log_2(8) = 3$ because $2^3 = 8$.

Q: What is the relationship between logarithmic functions and exponential functions?

A: Logarithmic functions and exponential functions are inverses of each other. This means that if $f(x) = 2^x$, then $f^{-1}(x) = \log_2(x)$.

Conclusion

In this article, we have answered some frequently asked questions about exponential equations and logarithmic functions. We hope that this article has provided you with a better understanding of these important mathematical concepts.

Key Takeaways

• Exponential equations are equations in which the variable appears as an exponent.
• Logarithmic functions are the inverse of exponential functions.
• The third logarithm law states that $\log(a) + \log(b) = \log(ab)$.
• Logarithmic functions and exponential functions are inverses of each other.

Further Reading

For further reading on exponential equations and logarithmic functions, we recommend the following resources:

• Exponential Equations
• Logarithmic Functions
• The Third Logarithm Law

Glossary

• Exponential Equation: An equation in which the variable appears as an exponent.
• Logarithmic Function: A function that is the inverse of an exponential function.
• Third Logarithm Law: A property of logarithms that states $\log(a) + \log(b) = \log(ab)$.

References

• [1] Exponential Equations
• [2] Logarithmic Functions
• [3] The Third Logarithm Law

About the Author

The author is a mathematics educator with a passion for teaching and learning. They have a strong background in mathematics and have taught a variety of courses, including algebra, geometry, and calculus.