Solving Exponential Equations With Natural Logarithms

Welcome! Let's dive into the world of exponential equations. This guide will walk you through solving an exponential equation, focusing on clarity and understanding. We will tackle the equation 5^(3x+8) = 13^(4-5x) step-by-step, finding both the exact solution using natural logarithms and an approximate solution rounded to four decimal places. This approach ensures you grasp the underlying concepts and gain the skills to solve similar problems confidently. So, grab your pen and paper, and let's get started!

Part a: Finding the Exact Solution with Natural Logarithms

Solving Exponential Equations with Natural Logarithms involves using the properties of logarithms to isolate the variable. The natural logarithm (ln) is particularly useful because it's the inverse function of the exponential function with base e. This means that ln(e^x) = x. To solve our equation, we'll apply the natural logarithm to both sides. This process helps us bring down the exponents, making it easier to manipulate the equation and solve for x. Remember, the goal is to isolate x to find its exact value.

Let's start with our equation: 5^(3x+8) = 13^(4-5x). The first step is to apply the natural logarithm (ln) to both sides of the equation. This gives us: ln(5^(3x+8)) = ln(13^(4-5x)). Next, we'll use the power rule of logarithms, which states that ln(a^b) = b * ln(a). Applying this rule to both sides, we get: (3x+8) * ln(5) = (4-5x) * ln(13). Now, we need to expand both sides of the equation by distributing the terms. This gives us: 3x * ln(5) + 8 * ln(5) = 4 * ln(13) - 5x * ln(13). The next step is to get all the terms containing x on one side of the equation and the constant terms on the other side. Let's move the -5x * ln(13) term to the left side and the 8 * ln(5) term to the right side. This gives us: 3x * ln(5) + 5x * ln(13) = 4 * ln(13) - 8 * ln(5). Now, we can factor out x from the left side of the equation: x * (3 * ln(5) + 5 * ln(13)) = 4 * ln(13) - 8 * ln(5). Finally, to isolate x, we divide both sides of the equation by (3 * ln(5) + 5 * ln(13)). This gives us the exact solution: x = (4 * ln(13) - 8 * ln(5)) / (3 * ln(5) + 5 * ln(13)). This is our exact answer expressed using natural logarithms. The ability to manipulate and simplify equations using logarithmic properties is crucial in various areas of mathematics and science.

Detailed Breakdown of the Logarithmic Transformation

Let's revisit the steps in detail to ensure complete clarity. Starting with 5^(3x+8) = 13^(4-5x), taking the natural logarithm of both sides is the first crucial step. It is mathematically valid and transforms the equation in a way that allows us to use the power rule. The power rule of logarithms ln(a^b) = b * ln(a) is then applied. This is where the exponents become coefficients. The power rule is a direct consequence of the properties of logarithms and exponents and is fundamental in simplifying logarithmic expressions. The distributive property is applied in expanding terms, ensuring each term is accounted for. Rearranging terms to isolate x is essential. This is a standard algebraic technique to group similar terms together. Factoring out x prepares us to solve for it directly. The final division isolates x, giving us the exact solution. This process demonstrates how logarithmic properties can be used to solve exponential equations, showing a clear pathway from the initial problem to the final solution. The correctness and efficiency of the calculation rely on the accurate application of the properties of logarithms. Each step must be executed carefully to minimize errors.

Part b: Approximating the Solution to Four Decimal Places

To find the Approximate Solution, we use the exact solution we found in part a: x = (4 * ln(13) - 8 * ln(5)) / (3 * ln(5) + 5 * ln(13)). We will now use a calculator to evaluate this expression and round the result to four decimal places. This gives us a numerical approximation of the solution. Using a calculator, we first calculate 4 * ln(13) ≈ 10.2315, 8 * ln(5) ≈ 12.8702, 3 * ln(5) ≈ 4.8283, and 5 * ln(13) ≈ 12.9244. Then we substitute these values into the equation: x ≈ (10.2315 - 12.8702) / (4.8283 + 12.9244). This simplifies to x ≈ -2.6387 / 17.7527. Performing the final division, we get x ≈ -0.1486. Therefore, the approximate solution, rounded to four decimal places, is x ≈ -0.1486. This numerical approximation provides a concrete value that can be easily understood and used in practical applications. Approximations are frequently used in scientific and engineering fields where exact values are often unnecessary, and a reasonable estimate is sufficient. The accuracy of the approximation depends on the precision of the calculations performed.

Step-by-Step Calculation with a Calculator

Let's break down the calculator steps for approximating the solution. First, input 4 * ln(13) and record the result (approximately 10.2315). Next, input 8 * ln(5) and record its value (approximately 12.8702). Subtract the second result from the first result: 10.2315 - 12.8702 ≈ -2.6387. Now, calculate the denominator. Input 3 * ln(5) and record it (approximately 4.8283). Then, calculate 5 * ln(13) and record its result (approximately 12.9244). Add these two results together: 4.8283 + 12.9244 ≈ 17.7527. Finally, divide the numerator (-2.6387) by the denominator (17.7527) to get the approximate solution: x ≈ -2.6387 / 17.7527 ≈ -0.1486. The use of a calculator helps to eliminate computational errors. The process requires precise input and careful recording of intermediate results to ensure accuracy. When using a calculator, make sure to use the natural logarithm function (usually labeled as