Solve The Exponential Equation E^(2x+3) = 7^(4/13). Express The Answer As An Exact Expression And A Decimal Approximation Rounded To Two Decimal Places.

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In the realm of mathematics, exponential equations hold a significant position, often appearing in various scientific and engineering disciplines. These equations, characterized by a variable in the exponent, can seem daunting at first glance. However, with a systematic approach and a clear understanding of the underlying principles, solving them becomes a manageable task. This article delves into the intricacies of solving exponential equations, providing a step-by-step guide illustrated with a detailed example. We will explore the necessary techniques, emphasizing both exact expressions and decimal approximations, offering a comprehensive understanding for anyone looking to master this crucial mathematical skill.

Exponential equations are equations in which the variable appears in the exponent. These equations are fundamental in modeling various real-world phenomena, such as population growth, radioactive decay, compound interest, and more. The general form of an exponential equation is often represented as:

af(x)=ba^{f(x)} = b

Where:

  • a is the base, a positive real number not equal to 1.
  • f(x) is the exponent, which is a function of the variable x.
  • b is a positive real number.

The key to solving exponential equations lies in understanding the properties of exponents and logarithms. Logarithms are the inverse functions of exponentials, and they provide a powerful tool for isolating the variable in the exponent. By applying logarithmic properties, we can transform complex exponential equations into simpler algebraic forms, making them easier to solve. Understanding the relationship between exponential and logarithmic functions is crucial for tackling various mathematical problems and real-world applications.

Let's tackle the exponential equation: $e^{2x+3} = 7^{\frac{4}{13}}$. This equation involves the natural exponential function, which has the base e (approximately 2.71828). The variable x appears in the exponent, making it an exponential equation. To solve this, we will follow a step-by-step process that leverages the properties of logarithms.

Step 1: Apply the Natural Logarithm

The first step in solving this exponential equation is to apply the natural logarithm (ln) to both sides of the equation. The natural logarithm is the logarithm to the base e, and it is particularly useful when dealing with exponential functions with base e. Applying the natural logarithm to both sides, we get:

ln(e2x+3)=ln(7413)ln(e^{2x+3}) = ln(7^{\frac{4}{13}})

This step is crucial because it allows us to use the property of logarithms that states $ln(a^b) = b \cdot ln(a)$. This property will help us bring the exponent down and isolate the variable x.

Step 2: Use the Power Rule of Logarithms

Next, we apply the power rule of logarithms to simplify the equation. The power rule states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In our case, we have $ln(e^{2x+3})$ on the left side and $ln(7^{\frac{4}{13}})$ on the right side. Applying the power rule, we get:

(2x+3)⋅ln(e)=413⋅ln(7)(2x+3) \cdot ln(e) = \frac{4}{13} \cdot ln(7)

Since the natural logarithm of e is 1 (i.e., $ln(e) = 1$), the left side simplifies further:

(2x+3)⋅1=413⋅ln(7)(2x+3) \cdot 1 = \frac{4}{13} \cdot ln(7)

2x+3=413⋅ln(7)2x+3 = \frac{4}{13} \cdot ln(7)

Step 3: Isolate the Variable x

Now, our goal is to isolate the variable x. To do this, we first subtract 3 from both sides of the equation:

2x=413⋅ln(7)−32x = \frac{4}{13} \cdot ln(7) - 3

Next, we divide both sides by 2 to solve for x:

x=12⋅(413⋅ln(7)−3)x = \frac{1}{2} \cdot (\frac{4}{13} \cdot ln(7) - 3)

This is the exact expression for the solution of the equation.

Step 4: Find the Exact Expression

The exact expression for x is:

x=12⋅(413⋅ln(7)−3)x = \frac{1}{2} \cdot (\frac{4}{13} \cdot ln(7) - 3)

This expression provides the precise value of x in terms of the natural logarithm of 7. It is often useful to have the exact expression for further calculations or theoretical analysis.

Step 5: Calculate the Decimal Approximation

To find the decimal approximation, we substitute the value of $ln(7)$ (approximately 1.9459) into the expression and perform the calculations:

x=12⋅(413⋅1.9459−3)x = \frac{1}{2} \cdot (\frac{4}{13} \cdot 1.9459 - 3)

First, we calculate $\frac{4}{13} \cdot 1.9459$:

413⋅1.9459≈0.5987\frac{4}{13} \cdot 1.9459 ≈ 0.5987

Then, we subtract 3:

0.5987−3=−2.40130.5987 - 3 = -2.4013

Finally, we multiply by $\frac{1}{2}$:

x≈12⋅(−2.4013)=−1.20065x ≈ \frac{1}{2} \cdot (-2.4013) = -1.20065

Rounding to two decimal places, we get:

x≈−1.20x ≈ -1.20

Therefore, the decimal approximation of the solution, rounded to two decimal places, is -1.20.

In summary, we have found both the exact expression and the decimal approximation for the solution of the exponential equation $e^{2x+3} = 7^{\frac{4}{13}}$. The exact expression is:

x=12⋅(413⋅ln(7)−3)x = \frac{1}{2} \cdot (\frac{4}{13} \cdot ln(7) - 3)

The decimal approximation, rounded to two decimal places, is:

x≈−1.20x ≈ -1.20

Another approach to solving this equation involves directly calculating the exponent on the right side and then applying the natural logarithm. Let's revisit the equation:

e2x+3=7413e^{2x+3} = 7^{\frac{4}{13}}

First, we can calculate $7^{\frac{4}{13}}$:

7413≈1.94727^{\frac{4}{13}} ≈ 1.9472

Now, the equation becomes:

e2x+3≈1.9472e^{2x+3} ≈ 1.9472

Next, we apply the natural logarithm to both sides:

ln(e2x+3)=ln(1.9472)ln(e^{2x+3}) = ln(1.9472)

Using the property of logarithms, we simplify the left side:

2x+3=ln(1.9472)2x+3 = ln(1.9472)

We calculate $ln(1.9472)$:

ln(1.9472)≈0.6665ln(1.9472) ≈ 0.6665

So, we have:

2x+3=0.66652x+3 = 0.6665

Now, we isolate x:

2x=0.6665−32x = 0.6665 - 3

2x=−2.33352x = -2.3335

x=−2.33352x = \frac{-2.3335}{2}

x≈−1.16675x ≈ -1.16675

Rounding to two decimal places, we get:

x≈−1.17x ≈ -1.17

This alternative method provides a slightly different decimal approximation due to rounding at intermediate steps. However, it demonstrates another valid approach to solving exponential equations.

When solving exponential equations, there are several common mistakes that students often make. Recognizing these pitfalls and understanding how to avoid them is crucial for achieving accurate solutions. Here are some of the most frequent errors and strategies to prevent them:

  1. Incorrect Application of Logarithm Properties:

    • Mistake: Applying the power rule incorrectly or misunderstanding the relationship between logarithms and exponents. For instance, incorrectly distributing a logarithm over a sum or difference.
    • How to Avoid: Thoroughly understand and memorize the properties of logarithms. Practice applying these properties in various scenarios. Double-check each step to ensure the properties are applied correctly.

  2. Rounding Errors:

    • Mistake: Rounding intermediate values too early in the calculation. This can lead to significant inaccuracies in the final answer.
    • How to Avoid: Keep intermediate values in their exact form as long as possible. If rounding is necessary, do it at the final step. Use a calculator with sufficient precision to minimize rounding errors.

  3. Forgetting the Base of the Logarithm:

    • Mistake: Neglecting to specify the base of the logarithm or using the wrong base. This is particularly relevant when dealing with different bases (e.g., natural logarithm (ln) vs. common logarithm (log)).
    • How to Avoid: Always explicitly write the base of the logarithm when necessary. Remember that the natural logarithm (ln) has a base of e, and the common logarithm (log) has a base of 10. Use the appropriate logarithm function based on the problem.

  4. Algebraic Errors:

    • Mistake: Making mistakes in algebraic manipulations, such as adding, subtracting, multiplying, or dividing terms. These errors can easily derail the solution process.
    • How to Avoid: Practice algebraic manipulations regularly. Work through each step carefully and double-check your work. Use parentheses to maintain the correct order of operations.

  5. Incorrectly Isolating the Variable:

    • Mistake: Failing to isolate the variable x correctly. This can involve improper use of inverse operations or errors in simplifying expressions.
    • How to Avoid: Follow the steps for isolating the variable systematically. Perform inverse operations on both sides of the equation to maintain balance. Double-check each step to ensure accuracy.

By being aware of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy and confidence in solving exponential equations. Practice and attention to detail are key to mastering this essential mathematical skill.

Exponential equations are not just abstract mathematical concepts; they have a wide range of real-world applications. Understanding these applications can provide a deeper appreciation for the importance of solving exponential equations. Here are some notable examples:

  1. Population Growth:

    • Application: Modeling the growth of populations (human, animal, or microbial) over time.
    • Explanation: Exponential equations can accurately describe population growth when resources are unlimited. The equation often takes the form $P(t) = P_0 \cdot e^{kt}$, where $P(t)$ is the population at time t, $P_0$ is the initial population, k is the growth rate, and e is the base of the natural logarithm. Solving exponential equations allows us to predict future population sizes or determine growth rates from historical data.

  2. Radioactive Decay:

    • Application: Determining the rate at which radioactive substances decay and calculating the remaining amount of a substance after a certain time.
    • Explanation: Radioactive decay follows an exponential decay model, described by the equation $N(t) = N_0 \cdot e^{-\lambda t}$, where $N(t)$ is the amount of the substance remaining at time t, $N_0$ is the initial amount, $\lambda$ is the decay constant, and e is the base of the natural logarithm. Solving exponential equations is essential in nuclear physics, archaeology (carbon dating), and medicine (radioactive isotopes in treatments).

  3. Compound Interest:

    • Application: Calculating the amount of money accumulated in an account with compound interest over time.
    • Explanation: Compound interest, where interest is earned on both the principal and accumulated interest, is modeled by an exponential equation. The formula is $A = P(1 + \frac{r}{n})^{nt}$, where A is the final amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. Solving exponential equations helps in financial planning, investment analysis, and understanding the effects of compounding over time.

  4. Chemical Reactions:

    • Application: Modeling the rates of chemical reactions, particularly first-order reactions where the rate of reaction is proportional to the concentration of one reactant.
    • Explanation: The concentration of a reactant in a first-order reaction decreases exponentially over time, described by the equation $A = [A]_0 \cdot e^{-kt}$, where $A$ is the concentration of reactant A at time t, $[A]_0$ is the initial concentration, and k is the rate constant. Solving exponential equations is crucial in chemical kinetics for determining reaction rates and predicting reactant concentrations.

  5. Cooling and Heating:

    • Application: Describing the temperature change of an object as it cools or heats up, known as Newton's Law of Cooling.
    • Explanation: Newton's Law of Cooling states that the rate of temperature change is proportional to the temperature difference between the object and its surroundings. This is modeled by an exponential equation: $T(t) = T_s + (T_0 - T_s)e^{-kt}$, where $T(t)$ is the temperature at time t, $T_s$ is the surrounding temperature, $T_0$ is the initial temperature, and k is a constant. Solving exponential equations helps in various applications, such as food processing, materials science, and climate modeling.

These examples illustrate the broad applicability of exponential equations in diverse fields. Mastering the techniques for solving them is invaluable for anyone pursuing studies or careers in science, engineering, finance, or other quantitative disciplines. The ability to model and analyze real-world phenomena using exponential equations provides a powerful tool for understanding and predicting future outcomes.

In this comprehensive guide, we've explored the process of solving exponential equations, starting from the fundamental principles and progressing to detailed step-by-step solutions. We’ve demonstrated how to solve the exponential equation $e^{2x+3} = 7^{\frac{4}{13}}$ by applying the natural logarithm, using the power rule of logarithms, and isolating the variable x. We've also emphasized the importance of expressing the answer in both exact form and as a decimal approximation, providing flexibility for different applications.

Furthermore, we addressed common mistakes that students often encounter when solving exponential equations, offering practical strategies to avoid these pitfalls. Understanding these common errors and how to prevent them is crucial for achieving accurate and reliable solutions. We also discussed an alternative method for solving the equation, which involves directly calculating the exponent and then applying the natural logarithm, demonstrating the versatility of different approaches.

Finally, we highlighted the real-world applications of exponential equations, illustrating their relevance in various fields such as population growth, radioactive decay, compound interest, chemical reactions, and cooling/heating processes. These applications underscore the practical significance of mastering exponential equations and their solutions.

By understanding the concepts, following the steps, and practicing regularly, anyone can develop the skills necessary to confidently solve exponential equations. This ability is not only valuable in academic settings but also in numerous professional and practical contexts, making the effort to master these techniques a worthwhile investment in one's mathematical toolkit.