Using the Property of Natural Logs

Oct 13, 2025 by recoveryshake.com 7 views

Introduction

Exponential functions are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore how to solve exponential functions using natural logs, a powerful technique that allows us to isolate the variable and find its value. We will use the given exponential function $e^{(3x - 10)} = 22.5$ as an example and walk through the steps to solve for $x$ .

What are Exponential Functions?

Exponential functions are functions of the form $f(x) = a^x$ , where $a$ is a positive constant and $x$ is the variable. These functions have a characteristic "S" shape, with the function increasing rapidly as $x$ increases. Exponential functions are used to model a wide range of phenomena, from population growth to chemical reactions.

The Role of Natural Logs

Natural logs, denoted by $\ln$ , are the inverse of exponential functions. They allow us to "undo" the exponential function and isolate the variable. In other words, if we have an exponential function $f(x) = a^x$ , we can take the natural log of both sides to get $x = \ln(f(x))$ . This is a powerful technique that allows us to solve exponential functions and find the value of the variable.

Step 1: Take the Natural Log of Both Sides

To solve the given exponential function $e^{(3x - 10)} = 22.5$ , we will take the natural log of both sides. This gives us:

\ln(e^{(3x - 10)}) = \ln(22.5)

Using the Property of Natural Logs

One of the key properties of natural logs is that they are the inverse of exponential functions. This means that we can "undo" the exponential function by taking the natural log of both sides. In other words, $\ln(e^x) = x$ . We can use this property to simplify the left-hand side of the equation:

3x - 10 = \ln(22.5)

Step 2: Add 10 to Both Sides

To isolate the term with the variable, we will add 10 to both sides of the equation:

3x = \ln(22.5) + 10

Step 3: Divide Both Sides by 3

Finally, we will divide both sides of the equation by 3 to solve for $x$ :

x = \frac{\ln(22.5) + 10}{3}

Rounding to the Nearest Tenth

We are asked to round our answer to the nearest tenth. To do this, we will use a calculator to find the value of $\ln(22.5)$ and then round the result to the nearest tenth:

x \approx \frac{3.10 + 10}{3} \approx \frac{13.10}{3} \approx 4.37

Conclusion

In this article, we have seen how to solve exponential functions using natural logs. We have walked through the steps to solve the given exponential function $e^{(3x - 10)} = 22.5$ and found the value of $x$ to be approximately 4.37. This technique is a powerful tool for solving exponential functions and finding the value of the variable.

Common Mistakes to Avoid

When solving exponential functions using natural logs, there are several common mistakes to avoid:

Not using the property of natural logs: Make sure to use the property of natural logs to simplify the left-hand side of the equation.
Not adding 10 to both sides: Make sure to add 10 to both sides of the equation to isolate the term with the variable.
Not dividing both sides by 3: Make sure to divide both sides of the equation by 3 to solve for $x$ .

Practice Problems

To practice solving exponential functions using natural logs, try the following problems:

$e^{(2x - 5)} = 15.6$
$e^{(x + 3)} = 20.2$
$e^{(4x - 2)} = 30.1$

Conclusion

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

A: Exponential functions and logarithmic functions are related but distinct concepts. Exponential functions are functions of the form $f(x) = a^x$ , where $a$ is a positive constant and $x$ is the variable. Logarithmic functions, on the other hand, are the inverse of exponential functions. They allow us to "undo" the exponential function and isolate the variable.

Q: Why do we use natural logs instead of other types of logs?

A: We use natural logs because they are the inverse of exponential functions. In other words, $\ln(e^x) = x$ . This property makes natural logs a powerful tool for solving exponential functions and finding the value of the variable.

Q: Can I use other types of logs, such as common logs or base-10 logs?

A: Yes, you can use other types of logs, but natural logs are the most commonly used and are the inverse of exponential functions. Common logs and base-10 logs are also useful, but they are not as widely used as natural logs.

Q: How do I know when to use natural logs to solve an exponential function?

A: You should use natural logs to solve an exponential function when the function is in the form $e^{(ax + b)} = c$ , where $a$ , $b$ , and $c$ are constants. This is because natural logs are the inverse of exponential functions, and they allow us to isolate the variable.

Q: What if the exponential function is not in the form $e^{(ax + b)} = c$ ?

A: If the exponential function is not in the form $e^{(ax + b)} = c$ , you may need to use other techniques, such as factoring or using the quadratic formula, to solve the function.

Q: Can I use natural logs to solve exponential functions with negative exponents?

A: Yes, you can use natural logs to solve exponential functions with negative exponents. To do this, you will need to use the property of natural logs that states $\ln(e^{-x}) = -x$ .

Q: How do I round my answer to the nearest tenth when solving an exponential function with natural logs?

A: To round your answer to the nearest tenth, you will need to use a calculator to find the value of the natural log and then round the result to the nearest tenth.

Q: What if I get a negative value for $x$ when solving an exponential function with natural logs?

A: If you get a negative value for $x$ when solving an exponential function with natural logs, you will need to check your work to make sure that you did not make a mistake. If you are sure that your work is correct, then the negative value for $x$ is a valid solution.

Q: Can I use natural logs to solve exponential functions with fractional exponents?

A: Yes, you can use natural logs to solve exponential functions with fractional exponents. To do this, you will need to use the property of natural logs that states $\ln(e^{(ax + b)}) = ax + b$ .

Q: How do I know if my solution is correct when solving an exponential function with natural logs?

A: To know if your solution is correct, you will need to check your work by plugging the value of $x$ back into the original equation and making sure that it is true. If the equation is true, then your solution is correct.

Conclusion

Solving exponential functions with natural logs is a powerful technique that allows us to isolate the variable and find its value. By following the steps outlined in this article and using the properties of natural logs, you can solve exponential functions and find the value of the variable. Remember to use the property of natural logs, add 10 to both sides, and divide both sides by 3 to solve for $x$ . With practice, you will become proficient in solving exponential functions using natural logs.