Solve The Equation. Give An Exact Solution, And Also Approximate The Solution To Four Decimal Places.$5^{x+2} = 7$Write The Exact Solution.$x = \square$ (Simplify Your Answer.)The Approximate Solution Is $\square$. (Do Not

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Introduction

In this article, we will delve into solving the equation $5^{x+2} = 7$ and find both the exact and approximate solutions. The exact solution will be expressed in its simplified form, while the approximate solution will be rounded to four decimal places. This problem involves logarithmic functions and their properties, which will be utilized to find the solutions.

Understanding the Equation

The given equation is $5^{x+2} = 7$. To solve for $x$, we need to isolate the variable. The first step is to apply the properties of exponents to simplify the equation. We can rewrite $5^{x+2}$ as $5^x \cdot 5^2$. This gives us the equation $5^x \cdot 25 = 7$.

Using Logarithms to Solve the Equation

To solve for $x$, we can use logarithms. We can take the logarithm of both sides of the equation to bring down the exponent. Let's use the natural logarithm (ln) for this purpose. Taking the natural logarithm of both sides gives us:

$$\ln(5^x \cdot 25) = \ln(7)$$

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

$$\ln(5^x) + \ln(25) = \ln(7)$$

Isolating the Variable

Now, we can isolate the variable $x$ by subtracting $\ln(25)$ from both sides of the equation:

$$\ln(5^x) = \ln(7) - \ln(25)$$

Applying the Exponential Function

To solve for $x$, we can apply the exponential function to both sides of the equation. This will bring down the exponent and give us:

$$5^x = e^{\ln(7) - \ln(25)}$$

Simplifying the Expression

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

$$5^x = \frac{7}{25}$$

Finding the Exact Solution

To find the exact solution, we can take the logarithm of both sides of the equation. Let's use the natural logarithm again:

$$\ln(5^x) = \ln\left(\frac{7}{25}\right)$$

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

$$x\ln(5) = \ln(7) - \ln(25)$$

Solving for x

Now, we can solve for $x$ by dividing both sides of the equation by $\ln(5)$:

$$x = \frac{\ln(7) - \ln(25)}{\ln(5)}$$

Simplifying the Expression

Using the property of logarithms that states $\ln(a/b) = \ln(a) - \ln(b)$, we can simplify the expression:

$$x = \frac{\ln(7) - \ln(25)}{\ln(5)}$$

Approximating the Solution

To approximate the solution, we can use a calculator to evaluate the expression:

$$x \approx \frac{\ln(7) - \ln(25)}{\ln(5)}$$

Using a calculator, we get:

$$x \approx -1.6094$$

Conclusion

In this article, we have solved the equation $5^{x+2} = 7$ and found both the exact and approximate solutions. The exact solution is $x = \frac{\ln(7) - \ln(25)}{\ln(5)}$, while the approximate solution is $x \approx -1.6094$. This problem involves logarithmic functions and their properties, which are essential tools in solving equations involving exponents.

Final Answer

The final answer is:

$x = \frac{\ln(7) - \ln(25)}{\ln(5)}$

$x \approx -1.6094$

Discussion

This problem is a classic example of how logarithmic functions can be used to solve equations involving exponents. The exact solution is expressed in its simplified form, while the approximate solution is rounded to four decimal places. This problem requires a good understanding of logarithmic functions and their properties, as well as the ability to apply them to solve equations.

Related Problems

• Solve the equation $2^{x+3} = 11$
• Find the exact and approximate solutions to the equation $3^{x+1} = 9$
• Use logarithmic functions to solve the equation $4^{x+2} = 16$

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Exponential Functions" by Math Open Reference
• [3] "Solving Equations Involving Exponents" by Khan Academy
  

Introduction

In the previous article, we solved the equation $5^{x+2} = 7$ and found both the exact and approximate solutions. In this article, we will answer some frequently asked questions related to solving equations involving exponents.

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

A: An exponential function is a function that involves an exponent, such as $f(x) = 2^x$. A logarithmic function is the inverse of an exponential function, such as $f(x) = \log_2(x)$.

Q: How do I solve an equation involving an exponential function?

A: To solve an equation involving an exponential function, you can use logarithmic functions to bring down the exponent. For example, if you have the equation $2^{x+3} = 11$, you can take the logarithm of both sides to get:

$$\ln(2^{x+3}) = \ln(11)$$

Q: What is the property of logarithms that states $\ln(ab) = \ln(a) + \ln(b)$?

A: This property is called the product rule of logarithms. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q: How do I simplify an expression involving logarithms?

A: To simplify an expression involving logarithms, you can use the properties of logarithms, such as the product rule and the quotient rule. For example, if you have the expression $\ln(7) - \ln(25)$, you can simplify it by using the quotient rule:

$$\ln(7) - \ln(25) = \ln\left(\frac{7}{25}\right)$$

Q: What is the difference between an exact solution and an approximate solution?

A: An exact solution is a solution that is expressed in its simplified form, without any approximations. An approximate solution is a solution that is rounded to a certain number of decimal places.

Q: How do I approximate a solution to an equation involving exponents?

A: To approximate a solution to an equation involving exponents, you can use a calculator to evaluate the expression. For example, if you have the equation $5^{x+2} = 7$, you can approximate the solution by using a calculator to evaluate the expression:

$$x \approx \frac{\ln(7) - \ln(25)}{\ln(5)}$$

Q: What are some common mistakes to avoid when solving equations involving exponents?

A: Some common mistakes to avoid when solving equations involving exponents include:

• Not using logarithmic functions to bring down the exponent
• Not simplifying the expression involving logarithms
• Not approximating the solution correctly
• Not checking the solution to make sure it is valid

Q: How do I check the solution to an equation involving exponents?

A: To check the solution to an equation involving exponents, you can plug the solution back into the original equation and make sure it is true. For example, if you have the equation $5^{x+2} = 7$ and you find the solution $x \approx -1.6094$, you can plug this value back into the original equation to check that it is true:

$$5^{(-1.6094)+2} \approx 7$$

Conclusion

In this article, we have answered some frequently asked questions related to solving equations involving exponents. We have discussed the properties of logarithmic functions, how to simplify expressions involving logarithms, and how to approximate solutions to equations involving exponents. We have also discussed some common mistakes to avoid when solving equations involving exponents and how to check the solution to make sure it is valid.

Final Answer

The final answer is:

• The difference between an exponential function and a logarithmic function is that an exponential function involves an exponent, while a logarithmic function is the inverse of an exponential function.
• To solve an equation involving an exponential function, you can use logarithmic functions to bring down the exponent.
• The property of logarithms that states $\ln(ab) = \ln(a) + \ln(b)$ is called the product rule of logarithms.
• To simplify an expression involving logarithms, you can use the properties of logarithms, such as the product rule and the quotient rule.
• An exact solution is a solution that is expressed in its simplified form, without any approximations, while an approximate solution is a solution that is rounded to a certain number of decimal places.
• To approximate a solution to an equation involving exponents, you can use a calculator to evaluate the expression.
• Some common mistakes to avoid when solving equations involving exponents include not using logarithmic functions to bring down the exponent, not simplifying the expression involving logarithms, not approximating the solution correctly, and not checking the solution to make sure it is valid.
• To check the solution to an equation involving exponents, you can plug the solution back into the original equation and make sure it is true.

Discussion

This article has provided a comprehensive overview of solving equations involving exponents. We have discussed the properties of logarithmic functions, how to simplify expressions involving logarithms, and how to approximate solutions to equations involving exponents. We have also discussed some common mistakes to avoid when solving equations involving exponents and how to check the solution to make sure it is valid.

Related Problems

• Solve the equation $2^{x+3} = 11$
• Find the exact and approximate solutions to the equation $3^{x+1} = 9$
• Use logarithmic functions to solve the equation $4^{x+2} = 16$

References

• [1] "Logarithmic Functions" by Math Open Reference
• [2] "Exponential Functions" by Math Open Reference
• [3] "Solving Equations Involving Exponents" by Khan Academy