How Can You Use A Logarithmic Function To Solve $8^x = 32,768$?Choose All The Statements That Apply:A. $8^x = 32,768$ Is Equivalent To $x = \frac{\log 32,768}{\log 8} = 5$.B. \$8^x = 32,768$[/tex\] Is

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Introduction

Exponential equations can be challenging to solve, especially when dealing with large numbers. However, with the help of logarithms, we can simplify these equations and find their solutions. In this article, we will explore how to use logarithmic functions to solve the equation $8^x = 32,768$.

Understanding Exponential Equations

An exponential equation is an equation in which the variable appears as an exponent. In the equation $8^x = 32,768$, the variable $x$ is the exponent of the base $8$. To solve this equation, we need to find the value of $x$ that makes the equation true.

Using Logarithms to Solve Exponential Equations

Logarithms are the inverse operation of exponentiation. In other words, if $a^x = b$, then $\log_a b = x$. We can use this property to solve exponential equations by taking the logarithm of both sides of the equation.

Step 1: Take the Logarithm of Both Sides

To solve the equation $8^x = 32,768$, we can take the logarithm of both sides of the equation. We will use the logarithm base $10$, but we can use any base we prefer.

log⁡108x=log⁡1032,768\log_{10} 8^x = \log_{10} 32,768

Step 2: Apply the Power Rule of Logarithms

The power rule of logarithms states that $\log_a b^c = c \log_a b$. We can apply this rule to the left-hand side of the equation.

xlog⁡108=log⁡1032,768x \log_{10} 8 = \log_{10} 32,768

Step 3: Divide Both Sides by the Logarithm of the Base

To isolate $x$, we can divide both sides of the equation by the logarithm of the base.

x=log⁡1032,768log⁡108x = \frac{\log_{10} 32,768}{\log_{10} 8}

Step 4: Evaluate the Logarithms

Now that we have isolated $x$, we can evaluate the logarithms.

x=log⁡1032,768log⁡108=4.5150.903≈5x = \frac{\log_{10} 32,768}{\log_{10} 8} = \frac{4.515}{0.903} \approx 5

Conclusion

In this article, we have shown how to use logarithmic functions to solve the equation $8^x = 32,768$. We took the logarithm of both sides of the equation, applied the power rule of logarithms, and divided both sides by the logarithm of the base to isolate $x$. We then evaluated the logarithms to find the solution.

Answer to the Discussion Category

A. $8^x = 32,768$ is equivalent to $x = \frac{\log 32,768}{\log 8} = 5$.

This statement is TRUE.

B. $8^x = 32,768$ is equivalent to $x = \frac{\log 32,768}{\log 8} = 4$.

This statement is FALSE.

Final Thoughts

Solving exponential equations with logarithms can be a powerful tool in mathematics. By following the steps outlined in this article, you can solve equations like $8^x = 32,768$ and gain a deeper understanding of logarithmic functions.

Additional Resources

For more information on logarithmic functions and exponential equations, check out the following resources:

  • Khan Academy: Logarithms
  • Mathway: Exponential Equations
  • Wolfram Alpha: Logarithmic Functions

References

  • "Logarithms" by Khan Academy
  • "Exponential Equations" by Mathway
  • "Logarithmic Functions" by Wolfram Alpha
    Frequently Asked Questions: Solving Exponential Equations with Logarithms ====================================================================

Q: What is the main concept behind solving exponential equations with logarithms?

A: The main concept behind solving exponential equations with logarithms is to use the property of logarithms that states $\log_a b^c = c \log_a b$. This allows us to rewrite the exponential equation in a form that can be solved using logarithms.

Q: How do I take the logarithm of both sides of an exponential equation?

A: To take the logarithm of both sides of an exponential equation, you can use the logarithm base of your choice. For example, if you have the equation $8^x = 32,768$, you can take the logarithm base 10 of both sides:

log⁡108x=log⁡1032,768\log_{10} 8^x = \log_{10} 32,768

Q: What is the power rule of logarithms, and how do I apply it?

A: The power rule of logarithms states that $\log_a b^c = c \log_a b$. To apply this rule, you can rewrite the exponential equation in a form that allows you to use the power rule. For example, if you have the equation $8^x = 32,768$, you can rewrite it as:

xlog⁡108=log⁡1032,768x \log_{10} 8 = \log_{10} 32,768

Q: How do I isolate the variable in an exponential equation using logarithms?

A: To isolate the variable in an exponential equation using logarithms, you can divide both sides of the equation by the logarithm of the base. For example, if you have the equation $x \log_{10} 8 = \log_{10} 32,768$, you can divide both sides by $\log_{10} 8$ to isolate $x$:

x=log⁡1032,768log⁡108x = \frac{\log_{10} 32,768}{\log_{10} 8}

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

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

  • Not using the correct base for the logarithm
  • Not applying the power rule of logarithms correctly
  • Not isolating the variable correctly
  • Not evaluating the logarithms correctly

Q: Can I use logarithms to solve exponential equations with bases other than 10?

A: Yes, you can use logarithms to solve exponential equations with bases other than 10. For example, if you have the equation $2^x = 64$, you can take the logarithm base 2 of both sides:

log⁡22x=log⁡264\log_2 2^x = \log_2 64

Q: How do I evaluate logarithms in exponential equations?

A: To evaluate logarithms in exponential equations, you can use a calculator or a logarithm table. For example, if you have the equation $x = \frac{\log_{10} 32,768}{\log_{10} 8}$, you can use a calculator to evaluate the logarithms:

x=4.5150.903≈5x = \frac{4.515}{0.903} \approx 5

Q: Can I use logarithms to solve exponential equations with negative exponents?

A: Yes, you can use logarithms to solve exponential equations with negative exponents. For example, if you have the equation $2^{-x} = 1/4$, you can take the logarithm base 2 of both sides:

log⁡22−x=log⁡21/4\log_2 2^{-x} = \log_2 1/4

Q: How do I apply logarithms to solve exponential equations with fractional exponents?

A: To apply logarithms to solve exponential equations with fractional exponents, you can use the property of logarithms that states $\log_a b^c = c \log_a b$. For example, if you have the equation $2^{3/4} = 8$, you can take the logarithm base 2 of both sides:

log⁡223/4=log⁡28\log_2 2^{3/4} = \log_2 8

Conclusion

In this article, we have answered some of the most frequently asked questions about solving exponential equations with logarithms. We have covered topics such as taking the logarithm of both sides of an exponential equation, applying the power rule of logarithms, isolating the variable, and evaluating logarithms. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in solving exponential equations with logarithms.