Solve For { X $}$ In The Equation: ${ 0.4^x = 0.064 }$

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Introduction

In this article, we will explore how to solve for $x$ in the equation $0.4^x = 0.064$. This equation involves an exponential term with a base of $0.4$ and an exponent of $x$. Our goal is to isolate the variable $x$ and find its value. We will use various mathematical techniques, including logarithmic properties and algebraic manipulations, to solve for $x$.

Understanding the Equation

The given equation is $0.4^x = 0.064$. This equation can be rewritten as $0.4^x = 64 \times 10^{-3}$. We can see that the base of the exponential term is $0.4$, and the exponent is $x$. Our objective is to find the value of $x$ that satisfies this equation.

Using Logarithms to Solve for $x$

One way to solve for $x$ is to use logarithms. We can take the logarithm of both sides of the equation to get:

$$\log(0.4^x) = \log(0.064)$$

Using the property of logarithms that states $\log(a^b) = b \log(a)$, we can rewrite the left-hand side of the equation as:

$$x \log(0.4) = \log(0.064)$$

Now, we can solve for $x$ by dividing both sides of the equation by $\log(0.4)$:

$$x = \frac{\log(0.064)}{\log(0.4)}$$

Evaluating the Expression

To find the value of $x$, we need to evaluate the expression $\frac{\log(0.064)}{\log(0.4)}$. We can use a calculator to find the values of the logarithms:

$$\log(0.064) \approx -1.795$$

$$\log(0.4) \approx -0.397$$

Now, we can substitute these values into the expression:

$$x \approx \frac{-1.795}{-0.397}$$

Calculating the Value of $x$

Using a calculator, we can evaluate the expression to find the value of $x$:

$$x \approx 4.52$$

Therefore, the value of $x$ that satisfies the equation $0.4^x = 0.064$ is approximately $4.52$.

Alternative Method: Using Exponential Properties

Another way to solve for $x$ is to use exponential properties. We can rewrite the equation as:

$$0.4^x = 64 \times 10^{-3}$$

Using the property of exponents that states $a^b = c \Rightarrow b = \log_a(c)$, we can rewrite the equation as:

$$x = \log_{0.4}(64 \times 10^{-3})$$

Evaluating the Expression

To find the value of $x$, we need to evaluate the expression $\log_{0.4}(64 \times 10^{-3})$. We can use a calculator to find the value of the logarithm:

$$\log_{0.4}(64 \times 10^{-3}) \approx 4.52$$

Therefore, the value of $x$ that satisfies the equation $0.4^x = 0.064$ is approximately $4.52$.

Conclusion

In this article, we have explored how to solve for $x$ in the equation $0.4^x = 0.064$. We have used logarithmic properties and algebraic manipulations to isolate the variable $x$ and find its value. We have also used exponential properties to solve for $x$. The value of $x$ that satisfies the equation is approximately $4.52$. This result can be verified using a calculator or by graphing the equation.

Frequently Asked Questions

• Q: What is the value of $x$ that satisfies the equation $0.4^x = 0.064$? A: The value of $x$ that satisfies the equation is approximately $4.52$.
• Q: How can I solve for $x$ in the equation $0.4^x = 0.064$? A: You can use logarithmic properties and algebraic manipulations to isolate the variable $x$ and find its value.
• Q: What is the base of the exponential term in the equation $0.4^x = 0.064$? A: The base of the exponential term is $0.4$.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Logarithmic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithmic.html

Further Reading

• "Exponential and Logarithmic Equations" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/exponential-logarithmic-equations.html
• "Solving Exponential and Logarithmic Equations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f2f7f/exponential-and-logarithmic-equations/v/solving-exponential-and-logarithmic-equations
  

Introduction

In our previous article, we explored how to solve for $x$ in the equation $0.4^x = 0.064$. We used logarithmic properties and algebraic manipulations to isolate the variable $x$ and find its value. In this article, we will answer some frequently asked questions about solving exponential equations.

Q&A

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

A: An exponential equation is an equation that involves an exponential term, such as $a^x = b$. A logarithmic equation is an equation that involves a logarithmic term, such as $\log_a(b) = c$. While both types of equations involve exponents, they are used to solve for different variables.

Q: How do I know which base to use when solving an exponential equation?

A: The base of the exponential term is usually given in the problem. If it is not given, you can try using different bases to see which one works. However, the most common bases are 2, 10, and e.

Q: Can I use logarithmic properties to solve any exponential equation?

A: No, not all exponential equations can be solved using logarithmic properties. You can only use logarithmic properties to solve exponential equations that have a base that is a positive number greater than 1.

Q: How do I know if an exponential equation has a solution?

A: An exponential equation has a solution if the base is a positive number greater than 1 and the exponent is a real number. If the base is a negative number or the exponent is not a real number, the equation may not have a solution.

Q: Can I use algebraic manipulations to solve an exponential equation?

A: Yes, you can use algebraic manipulations to solve an exponential equation. However, you need to be careful when using algebraic manipulations, as they can sometimes lead to incorrect solutions.

Q: How do I know if an exponential equation has a unique solution?

A: An exponential equation has a unique solution if the base is a positive number greater than 1 and the exponent is a real number. If the base is a negative number or the exponent is not a real number, the equation may not have a unique solution.

Q: Can I use a calculator to solve an exponential equation?

A: Yes, you can use a calculator to solve an exponential equation. However, you need to be careful when using a calculator, as it may not always give you the correct solution.

Q: How do I know if an exponential equation has a solution that is an integer?

A: An exponential equation has a solution that is an integer if the base is a positive number greater than 1 and the exponent is a positive integer. If the base is a negative number or the exponent is not a positive integer, the equation may not have a solution that is an integer.

Conclusion

In this article, we have answered some frequently asked questions about solving exponential equations. We have discussed the difference between exponential and logarithmic equations, how to choose a base, and how to use logarithmic properties and algebraic manipulations to solve exponential equations. We have also discussed how to determine if an exponential equation has a solution and how to use a calculator to solve an exponential equation.

Frequently Asked Questions

• Q: What is the difference between an exponential equation and a logarithmic equation? A: An exponential equation is an equation that involves an exponential term, such as $a^x = b$. A logarithmic equation is an equation that involves a logarithmic term, such as $\log_a(b) = c$.
• Q: How do I know which base to use when solving an exponential equation? A: The base of the exponential term is usually given in the problem. If it is not given, you can try using different bases to see which one works.
• Q: Can I use logarithmic properties to solve any exponential equation? A: No, not all exponential equations can be solved using logarithmic properties. You can only use logarithmic properties to solve exponential equations that have a base that is a positive number greater than 1.

References

• [1] "Exponential Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/exponential.html
• [2] "Logarithmic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithmic.html
• [3] "Solving Exponential and Logarithmic Equations" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f2f7f/exponential-and-logarithmic-equations/v/solving-exponential-and-logarithmic-equations

Further Reading

• "Exponential and Logarithmic Equations" by Math Is Fun. Retrieved from https://www.mathisfun.com/algebra/exponential-logarithmic-equations.html
• "Solving Exponential and Logarithmic Equations" by IXL. Retrieved from https://www.ixl.com/math/algebra/exponential-and-logarithmic-equations