Solving $4^{x+3} = 18$ First Steps And Detailed Solutions

Jul 7, 2025 by Admin 58 views

$Solving Exponential Equations: A Step-by-Step Guide to $4^{x+3} = 18$$

Introduction

In the realm of mathematics, solving exponential equations is a fundamental skill. Exponential equations, characterized by a variable in the exponent, often appear in various scientific and engineering applications. One common example is the equation $4^{x+3} = 18$ . Tackling such equations requires a strategic approach. In this comprehensive guide, we will explore the possible first steps in solving this equation, analyzing different methods and their effectiveness. Understanding the initial moves is crucial for simplifying the equation and paving the way for a complete solution. By dissecting the given options, we'll delve into the underlying principles of exponential functions and logarithms, providing a solid foundation for solving similar problems. This detailed exploration will not only benefit students learning algebra but also anyone looking to refresh their understanding of exponential equations.

Understanding the Equation

Before diving into specific methods, let's first understand the structure of the equation $4^{x+3} = 18$ . This is an exponential equation because the variable 'x' is part of the exponent. The base of the exponent is 4, and the exponent itself is 'x+3'. The equation states that 4 raised to the power of 'x+3' is equal to 18. To solve for 'x', we need to isolate the variable. This usually involves applying inverse operations and logarithmic properties. It's essential to recognize that direct algebraic manipulation, such as subtracting 18 from both sides, won't help isolate 'x'. Instead, we need to employ techniques that can "undo" the exponential operation. This is where logarithms come into play. Logarithms are the inverse functions of exponentials, and they provide a powerful tool for solving exponential equations. By applying logarithms, we can bring the exponent down and transform the equation into a more manageable form. This initial understanding sets the stage for evaluating the possible first steps and choosing the most efficient path to the solution.

Possible First Steps: An In-Depth Analysis

Now, let's analyze the suggested first steps in solving the equation $4^{x+3} = 18$ . Each option represents a different approach, and understanding their implications is crucial for choosing the most effective one. We'll dissect each option, explaining its rationale and potential benefits.

Option A: Rewrite each side with base 2

One possible first step is to rewrite each side of the equation with base 2. This is a viable strategy because 4 is a power of 2 ( $4 = 2^2$ ). Rewriting the left side with base 2 gives us $(2^2)^{x+3}$ . Using the power of a power rule, this simplifies to $2^{2(x+3)}$ . The equation then becomes $2^{2(x+3)} = 18$ . While the left side is now in base 2, the right side, 18, is not a direct power of 2. However, expressing both sides in terms of a common base can simplify the equation. This step can be seen as a way to prepare the equation for logarithmic operations. By having a common base, we can more easily apply logarithm properties later on. This approach aligns with the principle of simplifying expressions before applying more complex operations. While not a direct solution method, it sets the stage for further steps involving logarithms.

Option B: Subtract 18 from both sides

Another suggested step is to subtract 18 from both sides of the equation. This would result in $4^{x+3} - 18 = 0$ . While algebraically valid, this step doesn't bring us closer to isolating 'x'. Subtracting a constant from both sides doesn't address the exponential nature of the equation. The variable 'x' remains trapped in the exponent, and this operation doesn't provide a way to "free" it. In general, subtracting constants in exponential equations is rarely a useful initial step. The key to solving exponential equations lies in dealing with the exponent, and subtraction doesn't directly impact the exponential term. Therefore, while mathematically correct, this step is not a productive first step towards solving the equation.

Option C: Take the base-4 logarithm of each side

Taking the base-4 logarithm of each side is a highly effective first step. This is because the base of the exponential term is 4. Applying the base-4 logarithm directly "undoes" the exponentiation on the left side. When we take the base-4 logarithm of $4^{x+3}$ , we get $log_4(4^{x+3})$ . By the logarithmic property $log_b(b^x) = x$ , this simplifies to $x+3$ . The equation then becomes $x+3 = log_4(18)$ . This is a significant simplification, as 'x' is no longer in the exponent. The remaining step is to isolate 'x' by subtracting 3 from both sides: $x = log_4(18) - 3$ . This approach demonstrates the power of using the logarithm with the same base as the exponential term. It directly simplifies the equation and brings us closer to the solution. This method is a prime example of using inverse operations to solve equations.

Option D: Take the natural logarithm of each side

Taking the natural logarithm (ln) of each side is another valid first step. The natural logarithm is the logarithm with base 'e' (Euler's number, approximately 2.718). Applying the natural logarithm to both sides of the equation gives us $ln(4^{x+3}) = ln(18)$ . Using the power rule of logarithms, which states that $log_b(a^c) = c * log_b(a)$ , we can rewrite the left side as $(x+3)ln(4)$ . The equation now becomes $(x+3)ln(4) = ln(18)$ . This step is effective because it brings the exponent down as a coefficient. We can then isolate 'x' by dividing both sides by $ln(4)$ and subtracting 3. While the natural logarithm doesn't directly cancel out the base-4 exponential, it still allows us to manipulate the equation algebraically. This approach highlights the versatility of logarithms in solving exponential equations. The natural logarithm is particularly useful when dealing with exponential functions involving the natural base 'e', but it can be applied to any exponential equation.

Option E: Take the common logarithm of each side

Taking the common logarithm (log) of each side is also a legitimate initial step. The common logarithm is the logarithm with base 10. Applying the common logarithm to both sides of the equation $4^{x+3} = 18$ results in $log(4^{x+3}) = log(18)$ . Using the power rule of logarithms, we can rewrite the left side as $(x+3)log(4)$ . The equation then transforms into $(x+3)log(4) = log(18)$ . Similar to using the natural logarithm, this step brings the exponent down as a coefficient. We can then isolate 'x' by dividing both sides by $log(4)$ and subtracting 3. The common logarithm, like the natural logarithm, provides a way to transform the exponential equation into a linear equation. This approach demonstrates that multiple logarithmic bases can be used to solve the same equation. The choice of logarithm base often depends on personal preference or the availability of calculator functions. Both common and natural logarithms are standard tools in solving exponential problems.

Evaluating the Options and Choosing the Best First Step

After analyzing each option, we can now evaluate them in terms of their effectiveness as first steps. Some options lead to a more direct path to the solution, while others require additional steps or might not be as efficient.

Option A (Rewrite each side with base 2): This is a valid but not the most direct first step. While it simplifies the base on one side, it doesn't immediately isolate 'x' and requires further steps involving logarithms.
Option B (Subtract 18 from both sides): This is an incorrect first step. It doesn't address the exponential nature of the equation and doesn't help in isolating 'x'.
Option C (Take the base-4 logarithm of each side): This is an excellent first step. It directly simplifies the left side of the equation, making it easier to solve for 'x'.
Option D (Take the natural logarithm of each side): This is a good first step. It brings the exponent down as a coefficient and allows for algebraic manipulation to isolate 'x'.
Option E (Take the common logarithm of each side): This is also a good first step, similar to option D. It provides an alternative way to transform the equation using a different logarithmic base.

The best first step among these options is Option C: Take the base-4 logarithm of each side. This approach directly utilizes the inverse relationship between the base-4 exponential and the base-4 logarithm, leading to the most immediate simplification of the equation. Options D and E are also valid and will lead to the correct solution, but they require an extra step of dividing by $ln(4)$ or $log(4)$ , respectively. Option A, while mathematically sound, is less direct. Option B is ineffective as a first step.

Detailed Solution Using the Base-4 Logarithm

Let's demonstrate the complete solution using the base-4 logarithm, as it's the most efficient first step.

Original equation: $4^{x+3} = 18$
Take the base-4 logarithm of both sides: $log_4(4^{x+3}) = log_4(18)$
Simplify the left side using the logarithmic property $log_b(b^x) = x$ : $x + 3 = log_4(18)$
Isolate 'x' by subtracting 3 from both sides: $x = log_4(18) - 3$

At this point, we have an exact solution. To find an approximate decimal value, we can use the change of base formula for logarithms: $log_b(a) = \frac{log_c(a)}{log_c(b)}$ . We can change the base to either the common logarithm (base 10) or the natural logarithm (base e). Let's use the natural logarithm:

$log_4(18) = \frac{ln(18)}{ln(4)}$

Now we can substitute this back into our equation for 'x':

$x = \frac{ln(18)}{ln(4)} - 3$

Using a calculator, we find:

$ln(18) ≈ 2.8904$

$ln(4) ≈ 1.3863$

$\frac{ln(18)}{ln(4)} ≈ 2.08496$

$x ≈ 2.085 - 3$

$x ≈ -0.915$

Therefore, the solution to the equation $4^{x+3} = 18$ is approximately $x ≈ -0.915$ .

Alternative Solutions Using Natural and Common Logarithms

To further illustrate the versatility of logarithmic methods, let's solve the same equation using natural and common logarithms.

Solution using the Natural Logarithm

Original equation: $4^{x+3} = 18$
Take the natural logarithm of both sides: $ln(4^{x+3}) = ln(18)$
Apply the power rule of logarithms: $(x+3)ln(4) = ln(18)$
Divide both sides by $ln(4)$ : $x + 3 = \frac{ln(18)}{ln(4)}$
Isolate 'x' by subtracting 3 from both sides: $x = \frac{ln(18)}{ln(4)} - 3$

As we saw earlier, this leads to the same approximate solution:

$x ≈ -0.915$

Solution using the Common Logarithm

Original equation: $4^{x+3} = 18$
Take the common logarithm of both sides: $log(4^{x+3}) = log(18)$
Apply the power rule of logarithms: $(x+3)log(4) = log(18)$
Divide both sides by $log(4)$ : $x + 3 = \frac{log(18)}{log(4)}$
Isolate 'x' by subtracting 3 from both sides: $x = \frac{log(18)}{log(4)} - 3$

Using a calculator, we find:

$log(18) ≈ 1.2553$

$log(4) ≈ 0.6021$

$\frac{log(18)}{log(4)} ≈ 2.08496$

$x ≈ 2.085 - 3$

$x ≈ -0.915$

Again, we arrive at the same approximate solution, demonstrating that the choice of logarithm base doesn't affect the final result, as long as the logarithmic properties are applied correctly.

Key Takeaways and Best Practices for Solving Exponential Equations

Solving exponential equations involves a strategic application of logarithmic properties. Here are some key takeaways and best practices:

Understand the relationship between exponentials and logarithms: Logarithms are the inverse functions of exponentials, and they provide the key to solving exponential equations.
Choose the appropriate first step: Taking the logarithm with the same base as the exponential term is often the most efficient first step.
Apply logarithmic properties correctly: The power rule, product rule, and quotient rule of logarithms are essential for simplifying equations.
Isolate the variable: After applying logarithms, the goal is to isolate the variable using algebraic manipulations.
Use the change of base formula when needed: This formula allows you to convert logarithms from one base to another for calculator computations.
Check your solution: Always verify your solution by substituting it back into the original equation.

By following these guidelines, you can confidently approach and solve a wide range of exponential equations.

Conclusion

In conclusion, solving the exponential equation $4^{x+3} = 18$ involves applying logarithmic principles strategically. While multiple approaches are valid, taking the base-4 logarithm of each side presents the most direct path to the solution. Understanding the properties of logarithms and their inverse relationship with exponentials is crucial for success. Options such as taking the natural or common logarithm also lead to the correct answer, showcasing the flexibility of logarithmic methods. However, subtracting a constant from both sides is not a productive initial step. By mastering these techniques and best practices, you can confidently tackle exponential equations in various mathematical and scientific contexts. The key is to choose the method that best simplifies the equation and allows for the isolation of the variable, ultimately leading to an accurate solution.