Solving 3^(x+2) = 15 With The Change Of Base Formula

Introduction

In the realm of mathematics, solving exponential equations is a fundamental skill. These equations, where the variable appears in the exponent, often require specific techniques to isolate the unknown. One powerful method is the application of the change of base formula for logarithms. In this comprehensive guide, we will delve into the process of solving the exponential equation $3^{x+2} = 15$ using the change of base formula, $\log_b y = \frac{\log y}{\log b}$. We will break down each step with clarity, ensuring a thorough understanding for learners of all levels. This equation, $3^{x+2} = 15$, serves as an excellent example to illustrate the practical application of logarithmic properties and the change of base formula. Mastering this technique opens doors to solving a wide range of exponential problems encountered in various fields, including science, engineering, and finance. This article aims to provide not just the solution, but also a deep understanding of the underlying principles, empowering you to tackle similar problems with confidence. The change of base formula is a crucial tool in manipulating logarithms, particularly when dealing with bases that are not readily available on standard calculators. It allows us to convert logarithms from one base to another, typically to base 10 or base e, which are commonly supported on calculators. This transformation is essential for numerical evaluation and for simplifying expressions involving logarithms with different bases. The ability to navigate and apply this formula effectively is a testament to a strong foundation in logarithmic functions. As we progress through the solution, we will highlight the key properties of logarithms that come into play, reinforcing their importance in solving exponential equations. So, whether you are a student grappling with algebra or a professional seeking to refresh your mathematical skills, this guide offers a valuable resource for mastering the art of solving exponential equations using the change of base formula.

Understanding the Change of Base Formula

Before we tackle the equation $3^{x+2} = 15$, let's solidify our understanding of the change of base formula. This formula is the cornerstone of our solution, and a clear grasp of its mechanics is crucial. The change of base formula states that for any positive numbers b, y, and a (where b ≠ 1 and a ≠ 1), the logarithm of y to the base b can be expressed as the logarithm of y to the base a divided by the logarithm of b to the base a. Mathematically, this is represented as: $\log_b y = \frac{\log_a y}{\log_a b}$. The beauty of this formula lies in its flexibility. It allows us to switch between different logarithmic bases, which is particularly useful when dealing with bases that are not readily computable on standard calculators. Typically, we use base 10 (common logarithm) or base e (natural logarithm) as our new base, as these are standard functions available on most calculators. However, the formula holds true for any valid base a. To truly appreciate the power of the change of base formula, consider its applications beyond solving exponential equations. It's a fundamental tool in simplifying complex logarithmic expressions, comparing logarithms with different bases, and even in numerical computations where a specific base might be more convenient. For instance, in computer science, logarithms base 2 are frequently used, while in physics and engineering, natural logarithms (base e) often appear. The change of base formula allows us to bridge the gap between these different bases, making calculations and comparisons seamless. The intuition behind the formula stems from the fundamental definition of logarithms. Remember that $\log_b y$ represents the exponent to which we must raise b to obtain y. The change of base formula essentially breaks down this exponent into a ratio of exponents in a different base, making it easier to compute. By mastering this formula, you gain a deeper understanding of the nature of logarithms and their interrelationships, paving the way for tackling more advanced mathematical concepts. The formula also helps us to understand how logarithms with different bases are related to each other and how we can convert between them. This is a powerful tool in many areas of mathematics and science.

Applying Logarithms to Solve the Equation

Now, let's apply this powerful tool to solve the equation $3^{x+2} = 15$. Our goal is to isolate x, which currently resides in the exponent. To achieve this, we'll employ the fundamental property of logarithms that allows us to "bring down" the exponent. The first step involves taking the logarithm of both sides of the equation. This is a valid operation as long as we apply the same logarithmic function to both sides, maintaining the equality. We can choose any base for our logarithm, but for convenience, we'll use the common logarithm (base 10), denoted simply as "log". Applying the logarithm to both sides, we get: $\log(3^{x+2}) = \log(15)$. The next crucial step is to utilize the power rule of logarithms, which states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number. In mathematical terms, $\log(a^b) = b \cdot \log(a)$. Applying this rule to the left side of our equation, we get: $(x+2) \cdot \log(3) = \log(15)$. Notice how the exponent $(x+2)$ has now been brought down as a multiplier, bringing us closer to isolating x. This step is a direct application of the logarithmic property and is essential in solving exponential equations. The logarithmic function, in essence, acts as the inverse of the exponential function, allowing us to unravel the exponent and reveal the underlying variable. Now that we have $(x+2) \cdot \log(3) = \log(15)$, we can proceed to isolate $(x+2)$ by dividing both sides of the equation by $\log(3)$. This gives us: $x + 2 = \frac{\log(15)}{\log(3)}$. This is where the change of base formula's essence becomes evident. We have expressed the solution in terms of logarithms, and while we could directly compute $\log(15)$ and $\log(3)$ using a calculator, understanding the underlying structure is key. We now have an expression where x is almost isolated, and the final steps involve simple algebraic manipulation to obtain the numerical value of x. This process of applying logarithms and their properties is a fundamental technique in solving exponential equations and is widely applicable in various mathematical and scientific contexts. It showcases the power of logarithms in transforming complex exponential relationships into more manageable linear forms.

Isolating x and Applying the Change of Base Formula

Having arrived at the equation $x + 2 = \frac{\log(15)}{\log(3)}$, our next task is to isolate x. This involves a simple algebraic step: subtracting 2 from both sides of the equation. This gives us: $x = \frac{\log(15)}{\log(3)} - 2$. Now, we have x expressed in terms of logarithms, but to obtain a numerical value, we can utilize the change of base formula. Although it might seem like we've already implicitly used the change of base formula in the previous step (by expressing the solution in terms of base 10 logarithms), let's explicitly demonstrate its application here for clarity and deeper understanding. Recall the change of base formula: $\log_b y = \frac{\log_a y}{\log_a b}$. We can rewrite the term $\frac{\log(15)}{\log(3)}$ using a different base. While we've already used base 10, let's consider changing it back to base 3 for illustrative purposes. Applying the formula, we get: $\frac{\log(15)}{\log(3)} = \log_3(15)$. This step showcases the flexibility of the change of base formula – we can move between different bases as needed to simplify our expressions or gain a better understanding of the underlying relationships. Now our equation becomes: $x = \log_3(15) - 2$. At this point, we can either use a calculator that supports logarithms with arbitrary bases to directly compute $\log_3(15)$, or we can stick with base 10 logarithms and compute the value using the expression $\frac{\log(15)}{\log(3)}$. Both methods will yield the same result. The key takeaway here is that the change of base formula provides us with options. It allows us to express logarithms in a form that is most convenient for computation or analysis. Furthermore, understanding how to manipulate logarithmic expressions using this formula is crucial for tackling more complex logarithmic equations and inequalities. We've not only isolated x but also demonstrated how the change of base formula plays a pivotal role in simplifying and evaluating logarithmic terms. This process underscores the importance of mastering logarithmic properties and their applications in solving various mathematical problems. The change of base formula allows us to use a calculator to find the logarithm of any base.

Final Calculation and Solution

To complete the solution, we need to calculate the numerical value of $x = \log_3(15) - 2$. As discussed earlier, we can use a calculator that supports logarithms with arbitrary bases, or we can use the base 10 logarithms and compute $x = \frac{\log(15)}{\log(3)} - 2$. Let's proceed with the latter approach, as it's more commonly applicable since most calculators have built-in functions for base 10 logarithms. Using a calculator, we find that $\log(15) ≈ 1.1761$ and $\log(3) ≈ 0.4771$. Therefore, $\frac{\log(15)}{\log(3)} ≈ \frac{1.1761}{0.4771} ≈ 2.4649$. Now, substitute this value back into our equation for x: $x ≈ 2.4649 - 2$. This gives us: $x ≈ 0.4649$. Therefore, the solution to the equation $3^{x+2} = 15$ is approximately $x ≈ 0.4649$. It's always a good practice to verify our solution by plugging it back into the original equation. Substituting $x ≈ 0.4649$ into $3^{x+2}$, we get: $3^{0.4649+2} = 3^{2.4649} ≈ 14.999$, which is very close to 15. The slight discrepancy is due to rounding errors in our calculations. This verification step reinforces the accuracy of our solution and our understanding of the problem-solving process. We have successfully solved the exponential equation using logarithms and the change of base formula. This entire process highlights the power of logarithms in transforming exponential problems into more manageable algebraic ones. Moreover, it demonstrates the importance of mastering logarithmic properties and the change of base formula for tackling a wide range of mathematical challenges. The ability to solve such equations is not just a valuable skill in mathematics but also in various fields such as science, engineering, and finance, where exponential models are frequently encountered. By understanding the underlying principles and techniques, you are well-equipped to handle similar problems with confidence and accuracy. This detailed walkthrough has not only provided the solution but also the reasoning and methodology behind it, ensuring a comprehensive understanding of the subject matter.

Conclusion

In conclusion, we have successfully solved the exponential equation $3^{x+2} = 15$ using the change of base formula and the properties of logarithms. We began by understanding the change of base formula, $\log_b y = \frac{\log y}{\log b}$, which allows us to convert logarithms from one base to another. We then applied logarithms to both sides of the equation, utilized the power rule of logarithms to bring the exponent down, and isolated x. Finally, we used the change of base formula to evaluate the logarithmic terms and obtain a numerical solution. The approximate solution to the equation is $x ≈ 0.4649$. This exercise demonstrates the power and versatility of logarithms in solving exponential equations. The ability to manipulate logarithmic expressions, apply the change of base formula, and utilize logarithmic properties is a crucial skill in mathematics and various related fields. By mastering these techniques, you can confidently tackle a wide range of problems involving exponential and logarithmic functions. Moreover, this step-by-step guide has emphasized the importance of understanding the underlying principles and reasoning behind each step, ensuring a deeper and more lasting comprehension of the subject matter. The change of base formula is not just a tool for computation; it's a bridge connecting logarithms with different bases, enabling us to simplify expressions and solve equations more effectively. As you continue your mathematical journey, remember the power of logarithms and the change of base formula as valuable assets in your problem-solving toolkit. This example serves as a strong foundation for tackling more complex exponential and logarithmic problems, paving the way for further exploration and mastery of these fundamental mathematical concepts. The skills acquired through this process are transferable and applicable in various contexts, making this a worthwhile endeavor for any aspiring mathematician or scientist. The journey through solving this equation has not only provided a solution but also a deeper appreciation for the elegance and power of logarithmic functions.