Solve The Given Equation For $x$:$6^{4x-6} = 33$ X = X = X = [Insert Solution Here]You May Enter The Exact Value Or Round To 4 Decimal Places.

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of exponential functions and logarithms. In this article, we will focus on solving a specific exponential equation, $6^{4x-6} = 33$, and provide a step-by-step guide on how to find the value of $x$.

Understanding Exponential Equations

Exponential equations are equations that involve exponential functions, which are functions of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable. Exponential equations can be written in the form $a^x = b$, where $a$ and $b$ are constants.

Solving the Given Equation

To solve the given equation, $6^{4x-6} = 33$, we need to isolate the variable $x$. We can start by taking the logarithm of both sides of the equation. This will allow us to use the properties of logarithms to simplify the equation.

Taking the Logarithm of Both Sides

We can take the logarithm of both sides of the equation using any base. For simplicity, let's use the natural logarithm (base $e$). The equation becomes:

ln⁡(64x−6)=ln⁡(33)\ln(6^{4x-6}) = \ln(33)

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

(4x−6)ln⁡(6)=ln⁡(33)(4x-6)\ln(6) = \ln(33)

Isolating the Variable

Now, we can isolate the variable $x$ by dividing both sides of the equation by $\ln(6)$:

4x−6=ln⁡(33)ln⁡(6)4x-6 = \frac{\ln(33)}{\ln(6)}

Solving for x

Finally, we can solve for $x$ by adding $6$ to both sides of the equation and then dividing both sides by $4$:

x=ln⁡(33)ln⁡(6)+64x = \frac{\frac{\ln(33)}{\ln(6)} + 6}{4}

Evaluating the Expression

To find the value of $x$, we need to evaluate the expression:

x=ln⁡(33)ln⁡(6)+64x = \frac{\frac{\ln(33)}{\ln(6)} + 6}{4}

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

x≈3.713+64x \approx \frac{3.713 + 6}{4}

x≈9.7134x \approx \frac{9.713}{4}

x≈2.433x \approx 2.433

Conclusion

In this article, we solved the exponential equation $6^{4x-6} = 33$ using the properties of logarithms. We took the logarithm of both sides of the equation, isolated the variable $x$, and then solved for $x$. The value of $x$ is approximately $2.433$.

Frequently Asked Questions

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable using the properties of logarithms.

Q: What is the natural logarithm?

A: The natural logarithm is the logarithm with base $e$, where $e$ is a mathematical constant approximately equal to $2.718$.

References

  • [1] "Exponential Functions" by Math Is Fun
  • [2] "Logarithms" by Khan Academy
  • [3] "Exponential Equations" by Purplemath

Additional Resources

  • [1] "Exponential Functions" by Wolfram MathWorld
  • [2] "Logarithms" by Math Open Reference
  • [3] "Exponential Equations" by IXL Math

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Introduction

Exponential equations can be a challenging topic for many students and professionals. In this article, we will provide answers to some of the most frequently asked questions about exponential equations.

Q&A

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable using the properties of logarithms. This involves taking the logarithm of both sides of the equation and then using the properties of logarithms to simplify the equation.

Q: What is the natural logarithm?

A: The natural logarithm is the logarithm with base $e$, where $e$ is a mathematical constant approximately equal to $2.718$. The natural logarithm is denoted by the symbol $\ln(x)$.

Q: How do I use logarithms to solve exponential equations?

A: To use logarithms to solve exponential equations, you need to take the logarithm of both sides of the equation. This will allow you to use the properties of logarithms to simplify the equation and 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 using the correct base for the logarithm
  • Not simplifying the equation correctly
  • Not isolating the variable correctly
  • Not checking the solution for extraneous solutions

Q: How do I check my solution for extraneous solutions?

A: To check your solution for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and you need to find another solution.

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

A: Exponential equations have many real-world applications, including:

  • Modeling population growth
  • Modeling chemical reactions
  • Modeling financial growth
  • Modeling electrical circuits

Examples

Example 1: Solving an Exponential Equation

Solve the equation $2^x = 16$.

Solution

To solve this equation, we need to take the logarithm of both sides of the equation:

ln⁡(2x)=ln⁡(16)\ln(2^x) = \ln(16)

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

xln⁡(2)=ln⁡(16)x\ln(2) = \ln(16)

Now, we can isolate the variable $x$ by dividing both sides of the equation by $\ln(2)$:

x=ln⁡(16)ln⁡(2)x = \frac{\ln(16)}{\ln(2)}

Example 2: Checking for Extraneous Solutions

Solve the equation $3^x = 27$ and check for extraneous solutions.

Solution

To solve this equation, we need to take the logarithm of both sides of the equation:

ln⁡(3x)=ln⁡(27)\ln(3^x) = \ln(27)

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

xln⁡(3)=ln⁡(27)x\ln(3) = \ln(27)

Now, we can isolate the variable $x$ by dividing both sides of the equation by $\ln(3)$:

x=ln⁡(27)ln⁡(3)x = \frac{\ln(27)}{\ln(3)}

To check for extraneous solutions, we need to plug the solution back into the original equation:

3x=273^x = 27

3ln⁡(27)ln⁡(3)=273^{\frac{\ln(27)}{\ln(3)}} = 27

Using a calculator, we can find that the solution is true, so it is not an extraneous solution.

Conclusion

In this article, we provided answers to some of the most frequently asked questions about exponential equations. We also provided examples of how to solve exponential equations and how to check for extraneous solutions.

Frequently Asked Questions

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential function, which is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable using the properties of logarithms.

Q: What is the natural logarithm?

A: The natural logarithm is the logarithm with base $e$, where $e$ is a mathematical constant approximately equal to $2.718$.

References

  • [1] "Exponential Functions" by Math Is Fun
  • [2] "Logarithms" by Khan Academy
  • [3] "Exponential Equations" by Purplemath

Additional Resources

  • [1] "Exponential Functions" by Wolfram MathWorld
  • [2] "Logarithms" by Math Open Reference
  • [3] "Exponential Equations" by IXL Math