Solving 3^(x-4)=6 Using Change Of Base Formula A Step-by-Step Guide

In this comprehensive guide, we will delve into the process of solving the exponential equation $3^{x-4}=6$ using the change of base formula. This technique is a powerful tool in mathematics, particularly when dealing with logarithms and exponential functions. We'll break down each step, ensuring a clear understanding of the underlying principles. Our main focus will be on employing the change of base formula, which states that $\log_b y = \frac{\log y}{\log b}$. This formula allows us to convert logarithms from one base to another, making it easier to work with calculators that typically have built-in functions for common logarithms (base 10) and natural logarithms (base e).

Before we dive into the solution, let's take a moment to grasp the essence of the change of base formula. This formula is a cornerstone in logarithmic manipulations, providing a bridge between logarithms of different bases. The formula, mathematically expressed as $\log_b y = \frac{\log y}{\log b}$, tells us that the logarithm of y to the base b can be calculated by dividing the logarithm of y (in any base) by the logarithm of b (in the same base). Most commonly, we convert to base 10 (denoted as $\log$) or base e (denoted as $\ln$), as these are readily available on calculators. This transformation is crucial because it enables us to evaluate logarithms that our calculators might not directly compute. The beauty of this formula lies in its versatility; it allows us to choose a convenient base for computation, effectively bypassing the limitations of specific calculator functions. In the context of solving exponential equations, this formula proves invaluable when isolating the variable within the exponent, turning a seemingly complex problem into a manageable one.

Now, let’s apply this to our equation, $3^{x-4}=6$. Our primary goal is to isolate x, which is currently nestled within the exponent. To free x, we'll employ logarithms. The first step involves taking the logarithm of both sides of the equation. We can choose any base for the logarithm, but for simplicity and calculator compatibility, we'll use the common logarithm (base 10). This gives us $\log(3^{x-4}) = \log(6)$. The power rule of logarithms then comes into play, allowing us to bring the exponent $(x-4)$ down as a coefficient: $(x-4)\log(3) = \log(6)$.

Next, we aim to isolate $(x-4)$ by dividing both sides of the equation by $\log(3)$, resulting in $x-4 = \frac{\log(6)}{\log(3)}$. Now, we can use a calculator to find the values of $\log(6)$ and $\log(3)$. $\log(6)$ is approximately 0.778, and $\log(3)$ is approximately 0.477. Thus, we have $x-4 = \frac{0.778}{0.477}$, which simplifies to approximately 1.631. To finally solve for x, we add 4 to both sides of the equation: $x = 1.631 + 4$, giving us $x = 5.631$. Therefore, the solution to the equation $3^{x-4}=6$, rounded to the nearest thousandth, is 5.631. This step-by-step approach highlights the power of logarithms in unraveling exponential equations and underscores the significance of the change of base formula in making these calculations feasible.

When faced with an exponential equation like $3^{x-4} = 6$, a crucial step is to apply logarithms to both sides of the equation. This might seem like an arbitrary move, but it's a fundamental technique rooted in the properties of logarithms and exponents. The core idea is to leverage the inverse relationship between exponential functions and logarithmic functions. By taking the logarithm of both sides, we effectively "undo" the exponentiation, paving the way to isolate the variable x. The choice of the logarithmic base is flexible at this stage; however, using either the common logarithm (base 10) or the natural logarithm (base e) is generally preferred because most calculators have built-in functions for these bases. For instance, applying the common logarithm to both sides transforms our equation into $\log(3^{x-4}) = \log(6)$. This transformation is not just a mathematical trick; it's a strategic maneuver that allows us to bring the exponent down as a coefficient, simplifying the equation and making it solvable.

After applying logarithms to both sides of the equation, the next critical step is utilizing the power rule of logarithms. This rule is a cornerstone in logarithmic manipulations, and it's particularly handy when dealing with exponents. The power rule states that $\log_b(a^c) = c \log_b(a)$. In simpler terms, the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. This rule is precisely what we need to extract x from the exponent in our equation. Recall that we have $\log(3^{x-4}) = \log(6)$. Applying the power rule, we can rewrite the left side of the equation as $(x-4)\log(3)$. This transformation is significant because it brings the exponent $(x-4)$ down as a coefficient, effectively turning an exponential expression into a linear one. Our equation now looks like $(x-4)\log(3) = \log(6)$, which is much easier to handle algebraically. The power rule of logarithms acts as a bridge, connecting the world of exponents with the more manageable realm of linear equations, making it an indispensable tool in solving exponential problems.

With the power rule applied, the equation now reads $(x-4)\log(3) = \log(6)$. The next objective is to isolate the variable x. This involves a series of algebraic manipulations designed to get x by itself on one side of the equation. Our first step in this isolation process is to undo the multiplication by $\log(3)$. We achieve this by dividing both sides of the equation by $\log(3)$, resulting in $x-4 = \frac{\log(6)}{\log(3)}$. This division is a crucial step, as it separates the term containing x from the logarithmic terms. At this point, we've successfully isolated the $(x-4)$ term, bringing us closer to solving for x. The equation now presents a clear path forward: all that remains is to deal with the subtraction of 4. To completely isolate x, we need to add 4 to both sides of the equation. This final step will unveil the value of x, providing the solution to our original exponential equation. Isolating the variable is a fundamental skill in algebra, and in this context, it's the key to unlocking the value of x and solving the equation.

At this stage, our equation is $x-4 = \frac{\log(6)}{\log(3)}$. This is where the change of base formula and calculator evaluation come into play. While we could directly evaluate $\frac{\log(6)}{\log(3)}$ using a calculator, this fraction itself represents the logarithm of 6 to the base 3, according to the change of base formula. However, for practical purposes, we'll proceed with calculator evaluation. Using a calculator, we find that $\log(6)$ is approximately 0.778 and $\log(3)$ is approximately 0.477. Dividing these values gives us $\frac{0.778}{0.477} \approx 1.631$. Thus, our equation becomes $x-4 = 1.631$. This numerical approximation is a critical step in obtaining a concrete solution for x. It bridges the gap between the abstract logarithmic expression and a tangible decimal value. The use of a calculator at this juncture highlights the practical application of logarithmic principles in real-world calculations. By converting the logarithmic terms into decimal approximations, we pave the way for the final step in solving for x, which involves simple addition.

We've reached the final stretch in solving for x. Our equation stands at $x-4 = 1.631$. The last step is to find the final solution for x. To isolate x completely, we simply add 4 to both sides of the equation. This gives us $x = 1.631 + 4$, which simplifies to $x = 5.631$. Therefore, the solution to the equation $3^{x-4} = 6$, rounded to the nearest thousandth, is 5.631. This value represents the precise point where the exponential function $3^{x-4}$ intersects the horizontal line y = 6. This final calculation underscores the power of logarithms in solving exponential equations. By systematically applying logarithmic properties and algebraic manipulations, we've successfully untangled the equation and revealed the value of x. This process not only provides a numerical answer but also reinforces the interconnectedness of exponential and logarithmic functions in mathematics.

In conclusion, solving the exponential equation $3^{x-4}=6$ using the change of base formula demonstrates the elegance and power of logarithmic techniques. By taking the logarithm of both sides, applying the power rule, and utilizing the change of base formula (implicitly through calculator evaluation), we successfully isolated x and found the solution to be approximately 5.631. This step-by-step approach not only solves the specific problem but also provides a framework for tackling other exponential equations. The key takeaways include the importance of the change of base formula, the strategic use of logarithms to unravel exponents, and the systematic application of algebraic principles to isolate the variable. Mastering these techniques is crucial for anyone delving into advanced mathematical concepts and real-world applications involving exponential and logarithmic functions.