Solve Exponential Equation 11^(-x-5) = 18^(8x) With Logarithms

Introduction

In this comprehensive guide, we will delve into the process of solving exponential equations, focusing specifically on the equation 11^(-x-5) = 18^(8x). Exponential equations are a fundamental concept in mathematics, appearing in various fields such as physics, engineering, and finance. Mastering the techniques to solve these equations is crucial for any student or professional working with quantitative data. This article aims to provide a step-by-step approach, ensuring clarity and understanding for readers of all levels. We will explore the properties of logarithms and how they are instrumental in isolating the variable in exponential equations. Our discussion will cover the application of both base-10 and base-e logarithms, empowering you with the tools to tackle similar problems effectively. So, let's embark on this mathematical journey and unravel the solution to this intriguing equation.

The equation 11^(-x-5) = 18^(8x) presents an interesting challenge due to the variable x appearing in the exponent. To solve this, we need to employ the properties of logarithms. Logarithms are the inverse operation to exponentiation, and they allow us to "bring down" the exponent, making it easier to isolate the variable. The key idea here is to apply a logarithm to both sides of the equation. This maintains the equality while transforming the equation into a form that is solvable using algebraic manipulations. We will first discuss the general approach of using logarithms to solve exponential equations and then apply it specifically to our given equation. This approach involves taking the logarithm of both sides, using the power rule of logarithms to simplify, and then solving the resulting linear equation for x. Understanding this method will not only help you solve this particular problem but also equip you with the skills to solve a wide range of exponential equations. Let's dive into the specifics and break down each step to ensure a solid grasp of the process.

Understanding the properties of logarithms is essential when dealing with exponential equations. Logarithms have several key properties that make them incredibly useful for solving equations where the variable is in the exponent. The most relevant property for our problem is the power rule, which states that log_b(a^c) = c * log_b(a). This rule allows us to move the exponent c from the power to a coefficient, effectively "bringing down" the exponent. In the context of our equation, this means we can take the logarithm of both sides and then use the power rule to simplify the equation. Another important property is that logarithms are inverse functions of exponentials. This means that if b^y = x, then log_b(x) = y. We will use this relationship to rewrite the exponential equation in a logarithmic form, which will then allow us to isolate x. The choice of the base of the logarithm (whether base-10, base-e (natural logarithm), or any other base) does not affect the final solution, but it is important to be consistent throughout the process. We will demonstrate the solution using both base-10 and base-e logarithms to show that the result is the same regardless of the base used. With these logarithmic properties in mind, we can now proceed to the solution of the equation 11^(-x-5) = 18^(8x).

Step-by-Step Solution

1. Apply Logarithms to Both Sides

To begin solving the equation 11^(-x-5) = 18^(8x), the first crucial step involves applying a logarithm to both sides. This is a fundamental technique in solving exponential equations because it allows us to use the properties of logarithms to simplify the equation and isolate the variable x. We can choose either base-10 logarithms (log) or base-e logarithms (ln) for this purpose. The choice of base does not affect the final result, as long as we remain consistent throughout the process. Let's start by using the natural logarithm (ln) for this step, as it is commonly used and often simplifies calculations in calculus and other advanced mathematical fields. Applying the natural logarithm to both sides, we get:

ln(11^(-x-5)) = ln(18^(8x))

This step is essential because it transforms the exponential equation into a logarithmic equation, which we can then manipulate using the properties of logarithms. By taking the logarithm of both sides, we maintain the equality of the equation while setting the stage for the next step, where we will use the power rule of logarithms to further simplify the expression. The application of logarithms is a powerful tool in solving exponential equations, and it is the cornerstone of our approach to finding the solution for x in this equation. Now that we have applied the natural logarithm, we can proceed to the next step, which involves using the power rule to bring the exponents down.

2. Use the Power Rule of Logarithms

The next critical step in solving the equation ln(11^(-x-5)) = ln(18^(8x)) is to apply the power rule of logarithms. This rule states that log_b(a^c) = c * log_b(a), where b is the base of the logarithm, a is the argument, and c is the exponent. In our case, we have the natural logarithm (base e) applied to exponential expressions. Applying the power rule to both sides of the equation, we can rewrite it as follows:

(-x - 5) * ln(11) = (8x) * ln(18)

This transformation is significant because it removes the exponents and brings the variable x down as a coefficient, making the equation linear in terms of x. Now, we have a more manageable equation that we can solve using algebraic techniques. The power rule is a fundamental property of logarithms and is crucial for solving exponential equations. It allows us to manipulate the equation into a form where the variable can be isolated and solved for. With the exponents now removed, we can proceed to the next step, which involves expanding and rearranging the equation to group the terms containing x on one side and the constant terms on the other side. This will set us up to isolate x and find its value.

3. Expand and Rearrange the Equation

Following the application of the power rule, our equation is now in the form (-x - 5) * ln(11) = (8x) * ln(18). To solve for x, we need to expand the left side of the equation and then rearrange the terms so that all terms containing x are on one side and all constant terms are on the other side. Expanding the left side, we get:

-x * ln(11) - 5 * ln(11) = 8x * ln(18)

Now, we want to group the terms with x on one side and the constants on the other. To do this, we can add x * ln(11) to both sides and add 5 * ln(11) to both sides:

-5 * ln(11) = 8x * ln(18) + x * ln(11)

This step is crucial because it isolates the terms with x on one side, allowing us to factor out x in the next step. Rearranging the equation in this way is a standard algebraic technique for solving linear equations. By carefully moving the terms around, we are preparing the equation for the final steps of solving for x. With the terms now grouped appropriately, we can proceed to the next step, which involves factoring out x and then dividing to isolate it.

4. Factor out x

After rearranging the equation, we have -5 * ln(11) = 8x * ln(18) + x * ln(11). Now, we need to factor out x from the right side of the equation. This is a standard algebraic technique that allows us to isolate x and solve for its value. Factoring out x, we get:

-5 * ln(11) = x * (8 * ln(18) + ln(11))

This step is significant because it consolidates all the terms containing x into a single term, making it easier to isolate x. Factoring is a fundamental skill in algebra, and it is essential for solving equations of this type. By factoring out x, we have transformed the equation into a form where we can simply divide both sides by the expression in parentheses to solve for x. The next and final step will be to divide both sides by (8 * ln(18) + ln(11)) to isolate x and find the solution to the equation. With x now factored out, we are just one step away from the final answer.

5. Isolate x and Find the Solution

Following the factoring step, we have the equation -5 * ln(11) = x * (8 * ln(18) + ln(11)). To isolate x, we need to divide both sides of the equation by the expression in parentheses, which is (8 * ln(18) + ln(11)). This will give us the value of x. Performing the division, we get:

x = -5 * ln(11) / (8 * ln(18) + ln(11))

This is the exact solution for x in terms of natural logarithms. We can also express the solution using base-10 logarithms by applying the change of base formula, but the value will remain the same. This step is the culmination of all the previous steps, and it provides the final solution to the equation. By isolating x, we have successfully solved the exponential equation 11^(-x-5) = 18^(8x). The solution we obtained is an exact solution, meaning it is not an approximation. It is expressed in terms of logarithms, which is the most precise way to represent the solution. Now, let's calculate the approximate numerical value of x to get a better understanding of its magnitude.

6. Calculate the Approximate Value

To obtain an approximate numerical value for x, we can use a calculator to evaluate the expression we found in the previous step: x = -5 * ln(11) / (8 * ln(18) + ln(11)). Using a calculator, we find that:

ln(11) ≈ 2.3979

ln(18) ≈ 2.8904

Now, substitute these values into the expression for x:

x ≈ -5 * 2.3979 / (8 * 2.8904 + 2.3979)

x ≈ -11.9895 / (23.1232 + 2.3979)

x ≈ -11.9895 / 25.5211

x ≈ -0.4698

Therefore, the approximate value of x is -0.4698. This value provides a numerical approximation of the exact solution we found earlier. It is important to note that this is an approximation, and the exact solution is given by the expression involving logarithms. However, the approximate value helps us understand the magnitude and sign of the solution. In this case, x is approximately -0.4698, which is a negative number less than 1 in absolute value. This completes the solution process for the equation 11^(-x-5) = 18^(8x). We have found both the exact solution in terms of logarithms and an approximate numerical value.

Verification of the Solution

To ensure the accuracy of our solution, it is crucial to verify the result by substituting the approximate value of x back into the original equation, 11^(-x-5) = 18^(8x). If our solution is correct, the left-hand side (LHS) of the equation should be approximately equal to the right-hand side (RHS) when we substitute x ≈ -0.4698. Let's perform this verification step by step. First, substitute x into the LHS:

LHS = 11^(-(-0.4698)-5)

LHS = 11^(0.4698-5)

LHS = 11^(-4.5302)

Now, calculate the value:

LHS ≈ 11^(-4.5302) ≈ 0.0000486

Next, substitute x into the RHS:

RHS = 18^(8*(-0.4698))

RHS = 18^(-3.7584)

Now, calculate the value:

RHS ≈ 18^(-3.7584) ≈ 0.0000485

Comparing the LHS and RHS values, we can see that they are very close to each other (0.0000486 ≈ 0.0000485). The slight difference is due to rounding errors in our approximation of x and the logarithmic values. However, the fact that the LHS and RHS are so close confirms that our solution is correct. This verification step is an important part of the problem-solving process, as it helps us catch any errors and ensure the accuracy of our results. By substituting the approximate value of x back into the original equation, we have gained confidence that our solution is indeed correct.

Conclusion

In conclusion, we have successfully solved the exponential equation 11^(-x-5) = 18^(8x). We started by applying logarithms to both sides of the equation, which allowed us to use the power rule of logarithms to simplify the expression. We then expanded and rearranged the equation to isolate the terms containing x. By factoring out x and dividing, we found the exact solution in terms of natural logarithms: x = -5 * ln(11) / (8 * ln(18) + ln(11)). Furthermore, we calculated an approximate numerical value for x, which is approximately -0.4698. To verify our solution, we substituted the approximate value back into the original equation and confirmed that the left-hand side was approximately equal to the right-hand side. This step-by-step process demonstrates a systematic approach to solving exponential equations, which can be applied to a wide range of similar problems.

Understanding the properties of logarithms and their application in solving exponential equations is a valuable skill in mathematics and various scientific fields. The techniques we have discussed in this article provide a solid foundation for tackling more complex equations and mathematical problems. By mastering these concepts, you can confidently approach exponential equations and find accurate solutions. The ability to solve such equations is not only useful in academic settings but also in practical applications such as financial modeling, population growth analysis, and radioactive decay calculations. Therefore, the knowledge and skills gained from this guide will serve you well in your future mathematical endeavors.

Repair Input Keyword

Solve the equation 11^(-x-5) = 18^(8x) for x. Find the exact answer using logarithms (base-10 or base-e).