Solve The Equations Using Exponential Or Logarithmic Equations.11. $\log (5x - 2) = 2$12. $4^{(x+2)} - 16 = 60$13. $e^x = 7$

Introduction

In this article, we will explore three different equations that can be solved using exponential or logarithmic equations. These types of equations are commonly used in mathematics and are essential for solving various problems in fields such as physics, engineering, and economics. We will start by solving the equation $\log (5x - 2) = 2$, followed by the equation $4^{(x+2)} - 16 = 60$, and finally, the equation $e^x = 7$.

Equation 1: $\log (5x - 2) = 2$

Step 1: Understanding the Equation

The given equation is $\log (5x - 2) = 2$. This is a logarithmic equation, where the logarithm of an expression is equal to a certain value. To solve this equation, we need to get rid of the logarithm and isolate the variable $x$.

Step 2: Converting to Exponential Form

To get rid of the logarithm, we can convert the equation to exponential form. The exponential form of a logarithmic equation is $a^{\log_a(x)} = x$. In this case, we can rewrite the equation as $10^2 = 5x - 2$.

Step 3: Simplifying the Equation

Now, we can simplify the equation by evaluating the exponential expression. $10^2 = 100$, so the equation becomes $100 = 5x - 2$.

Step 4: Solving for $x$

To solve for $x$, we need to isolate the variable. We can do this by adding $2$ to both sides of the equation, which gives us $102 = 5x$. Then, we can divide both sides by $5$ to get $x = 20.4$.

Conclusion

The solution to the equation $\log (5x - 2) = 2$ is $x = 20.4$.

Equation 2: $4^{(x+2)} - 16 = 60$

Step 1: Understanding the Equation

The given equation is $4^{(x+2)} - 16 = 60$. This is an exponential equation, where the base is $4$ and the exponent is $(x+2)$. To solve this equation, we need to get rid of the exponent and isolate the variable $x$.

Step 2: Adding $16$ to Both Sides

To get rid of the constant term, we can add $16$ to both sides of the equation, which gives us $4^{(x+2)} = 76$.

Step 3: Taking the Logarithm of Both Sides

To get rid of the exponent, we can take the logarithm of both sides of the equation. We can use any base for the logarithm, but let's use the natural logarithm. This gives us $\ln(4^{(x+2)}) = \ln(76)$.

Step 4: Using the Power Rule of Logarithms

The power rule of logarithms states that $\ln(a^b) = b\ln(a)$. We can use this rule to rewrite the equation as $(x+2)\ln(4) = \ln(76)$.

Step 5: Solving for $x$

To solve for $x$, we need to isolate the variable. We can do this by dividing both sides of the equation by $\ln(4)$, which gives us $(x+2) = \frac{\ln(76)}{\ln(4)}$. Then, we can subtract $2$ from both sides to get $x = \frac{\ln(76)}{\ln(4)} - 2$.

Conclusion

The solution to the equation $4^{(x+2)} - 16 = 60$ is $x = \frac{\ln(76)}{\ln(4)} - 2$.

Equation 3: $e^x = 7$

Step 1: Understanding the Equation

The given equation is $e^x = 7$. This is an exponential equation, where the base is $e$ and the exponent is $x$. To solve this equation, we need to get rid of the exponent and isolate the variable $x$.

Step 2: Taking the Natural Logarithm of Both Sides

To get rid of the exponent, we can take the natural logarithm of both sides of the equation. This gives us $\ln(e^x) = \ln(7)$.

Step 3: Using the Power Rule of Logarithms

The power rule of logarithms states that $\ln(a^b) = b\ln(a)$. We can use this rule to rewrite the equation as $x\ln(e) = \ln(7)$.

Step 4: Simplifying the Equation

Since $\ln(e) = 1$, we can simplify the equation to $x = \ln(7)$.

Conclusion

The solution to the equation $e^x = 7$ is $x = \ln(7)$.

Conclusion

$$$$$$$$

Introduction

In our previous article, we explored three different equations that can be solved using exponential or logarithmic equations. In this article, we will provide a Q&A section to help you better understand the concepts and techniques used to solve these equations.

Q: What is the difference between exponential and logarithmic equations?

A: Exponential equations are equations that involve an exponential expression, where the base is raised to a certain power. Logarithmic equations, on the other hand, are equations that involve a logarithmic expression, where the logarithm of an expression is equal to a certain value.

Q: How do I convert a logarithmic equation to exponential form?

A: To convert a logarithmic equation to exponential form, you can use the following formula: $a^{\log_a(x)} = x$. For example, if you have the equation $\log (5x - 2) = 2$, you can convert it to exponential form by rewriting it as $10^2 = 5x - 2$.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that $\ln(a^b) = b\ln(a)$. This rule allows you to rewrite an exponential expression in terms of a logarithmic expression.

Q: How do I use the power rule of logarithms to solve an equation?

A: To use the power rule of logarithms to solve an equation, you can take the logarithm of both sides of the equation and then use the power rule to rewrite the equation in a simpler form. For example, if you have the equation $4^{(x+2)} - 16 = 60$, you can take the logarithm of both sides and then use the power rule to rewrite the equation as $(x+2)\ln(4) = \ln(76)$.

Q: What is the natural logarithm?

A: The natural logarithm is a logarithmic function that is based on the number $e$. It is denoted by the symbol $\ln(x)$ and is defined as the logarithm of $x$ to the base $e$.

Q: How do I use the natural logarithm to solve an equation?

A: To use the natural logarithm to solve an equation, you can take the natural logarithm of both sides of the equation and then use the properties of logarithms to simplify the equation. For example, if you have the equation $e^x = 7$, you can take the natural logarithm of both sides and then simplify the equation to $x = \ln(7)$.

Q: What are some common mistakes to avoid when solving exponential or logarithmic equations?

A: Some common mistakes to avoid when solving exponential or logarithmic equations include:

• Not checking the domain of the equation
• Not using the correct base for the logarithm
• Not simplifying the equation correctly
• Not checking the solution for extraneous solutions

Conclusion

In this article, we have provided a Q&A section to help you better understand the concepts and techniques used to solve exponential or logarithmic equations. We hope that this article has been helpful in clarifying any questions you may have had about these types of equations.

Additional Resources

If you are looking for additional resources to help you learn more about exponential or logarithmic equations, we recommend the following:

• Khan Academy: Exponential and Logarithmic Equations
• Mathway: Exponential and Logarithmic Equations
• Wolfram Alpha: Exponential and Logarithmic Equations

We hope that this article has been helpful in your studies. If you have any further questions or need additional help, please don't hesitate to ask.