Match The Equation 4 2 X − 5 = 6 4^{2x-5} = 6 4 2 X − 5 = 6 With The Corresponding Graph.

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Introduction

Exponential equations and functions are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and finance. In this article, we will focus on solving the equation 42x−5=64^{2x-5} = 6 and matching it with the corresponding graph.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential function, which is a function of the form f(x)=axf(x) = a^x, where aa is a positive real number. The equation 42x−5=64^{2x-5} = 6 is an example of an exponential equation, where the base is 44 and the exponent is 2x−52x-5.

Solving Exponential Equations

To solve the equation 42x−5=64^{2x-5} = 6, we need to isolate the variable xx. We can start by taking the logarithm of both sides of the equation. The logarithm of a number is the power to which a base must be raised to produce that number. In this case, we can use the logarithm base 44 to simplify the equation.

Using Logarithms to Solve the Equation

We can rewrite the equation 42x−5=64^{2x-5} = 6 as:

log⁡4(42x−5)=log⁡4(6)\log_4(4^{2x-5}) = \log_4(6)

Using the property of logarithms that log⁡a(ax)=x\log_a(a^x) = x, we can simplify the left-hand side of the equation:

2x−5=log⁡4(6)2x-5 = \log_4(6)

Now, we can solve for xx by isolating it on one side of the equation:

2x=log⁡4(6)+52x = \log_4(6) + 5

x=log⁡4(6)+52x = \frac{\log_4(6) + 5}{2}

Evaluating the Logarithm

To evaluate the logarithm log⁡4(6)\log_4(6), we can use the change of base formula, which states that log⁡a(b)=log⁡c(b)log⁡c(a)\log_a(b) = \frac{\log_c(b)}{\log_c(a)} for any positive real numbers aa, bb, and cc. In this case, we can choose base 1010 as the new base:

log⁡4(6)=log⁡10(6)log⁡10(4)\log_4(6) = \frac{\log_{10}(6)}{\log_{10}(4)}

Using a calculator to evaluate the logarithms, we get:

log⁡4(6)≈0.77820.6021≈1.294\log_4(6) \approx \frac{0.7782}{0.6021} \approx 1.294

Solving for x

Now that we have evaluated the logarithm, we can substitute the value back into the equation for xx:

x=1.294+52x = \frac{1.294 + 5}{2}

x≈6.2942x \approx \frac{6.294}{2}

x≈3.147x \approx 3.147

Graphing Exponential Functions

Now that we have solved the equation 42x−5=64^{2x-5} = 6, we can graph the corresponding exponential function. The general form of an exponential function is f(x)=axf(x) = a^x, where aa is a positive real number. In this case, the base is 44, so the function is f(x)=4xf(x) = 4^x.

Graphing the Function

To graph the function f(x)=4xf(x) = 4^x, we can use a graphing calculator or a computer algebra system. The graph of the function is a curve that passes through the point (0,1)(0, 1), since 40=14^0 = 1. The graph also passes through the point (1,4)(1, 4), since 41=44^1 = 4.

Matching the Graph with the Equation

Now that we have graphed the function f(x)=4xf(x) = 4^x, we can match it with the equation 42x−5=64^{2x-5} = 6. The graph of the function passes through the point (3.147,6)(3.147, 6), which corresponds to the solution we found earlier.

Conclusion

In this article, we have solved the equation 42x−5=64^{2x-5} = 6 and matched it with the corresponding graph. We used logarithms to simplify the equation and solve for xx. We also graphed the exponential function f(x)=4xf(x) = 4^x and matched it with the equation. The graph of the function passes through the point (3.147,6)(3.147, 6), which corresponds to the solution we found earlier.

Applications of Exponential Equations

Exponential equations have many applications in science, engineering, and finance. For example, they can be used to model population growth, chemical reactions, and financial investments. In this article, we have focused on solving a simple exponential equation and matching it with the corresponding graph. However, there are many more complex exponential equations that can be solved using logarithms and other mathematical techniques.

Future Directions

In the future, we can explore more complex exponential equations and their applications in science, engineering, and finance. We can also use computer algebra systems and graphing calculators to visualize the graphs of exponential functions and match them with the corresponding equations.

References

  • [1] "Exponential Functions" by Math Open Reference
  • [2] "Logarithms" by Math Is Fun
  • [3] "Graphing Exponential Functions" by Khan Academy

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of sources.

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Introduction

In our previous article, we explored the concept of exponential equations and graphs, and how to solve them using logarithms. In this article, we will provide a Q&A guide to help you better understand the topic.

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)=axf(x) = a^x, where aa is a positive real number.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use logarithms to simplify the equation and isolate the variable. You can use the change of base formula to evaluate the logarithm, and then solve for the variable.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to change the base of a logarithm. It is given by:

log⁡a(b)=log⁡c(b)log⁡c(a)\log_a(b) = \frac{\log_c(b)}{\log_c(a)}

where aa, bb, and cc are positive real numbers.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer algebra system. The graph of an exponential function is a curve that passes through the point (0,1)(0, 1), since a0=1a^0 = 1.

Q: What is the significance of the point (0, 1) in an exponential graph?

A: The point (0,1)(0, 1) is significant in an exponential graph because it represents the value of the function when the input is 00. Since a0=1a^0 = 1, the graph of an exponential function always passes through the point (0,1)(0, 1).

Q: How do I match an exponential graph with an equation?

A: To match an exponential graph with an equation, you can use the point on the graph that corresponds to the solution of the equation. For example, if the equation is 42x−5=64^{2x-5} = 6, you can find the point on the graph that corresponds to the solution x=3.147x = 3.147.

Q: What are some common applications of exponential equations?

A: Exponential equations have many applications in science, engineering, and finance. Some common applications include:

  • Modeling population growth
  • Chemical reactions
  • Financial investments
  • Electrical circuits

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

A: To use logarithms to solve exponential equations, you can take the logarithm of both sides of the equation and then use the change of base formula to evaluate the logarithm. You can then solve for the variable.

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

A: A logarithmic function is a function of the form f(x)=log⁡a(x)f(x) = \log_a(x), where aa is a positive real number. An exponential function is a function of the form f(x)=axf(x) = a^x, where aa is a positive real number. While logarithmic and exponential functions are related, they are not the same.

Q: How do I choose the base of a logarithm?

A: When choosing the base of a logarithm, you should choose a base that is convenient for the problem you are working on. For example, if you are working with a problem that involves base 1010, it may be easier to use base 1010 as the base of the logarithm.

Conclusion

In this article, we have provided a Q&A guide to help you better understand exponential equations and graphs. We have covered topics such as solving exponential equations, graphing exponential functions, and matching exponential graphs with equations. We hope that this guide has been helpful in your studies.

References

  • [1] "Exponential Functions" by Math Open Reference
  • [2] "Logarithms" by Math Is Fun
  • [3] "Graphing Exponential Functions" by Khan Academy

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of sources.