Solve Exponential Equations Using Logarithms A Step By Step Guide

Introduction

In mathematics, exponential equations are those in which the variable appears in the exponent. Solving these equations often requires the use of logarithms, which are the inverse operation to exponentiation. This article will delve into the process of using logarithms to solve exponential equations, providing a detailed explanation and step-by-step approach to tackle such problems. We will focus on an example equation, $4^{x+3} = 7^{x-6}$, to illustrate the method. Understanding how to solve exponential equations is crucial in various fields, including science, engineering, and finance, where exponential models are frequently used to describe growth and decay phenomena.

Understanding Exponential Equations

Exponential equations are of the form $a^{f(x)} = b^{g(x)}$, where $a$ and $b$ are constants and $f(x)$ and $g(x)$ are functions of $x$. The key to solving these equations lies in the properties of logarithms. A logarithm is the inverse operation to exponentiation, meaning that if $a^y = x$, then $\log_a(x) = y$. The most commonly used logarithms are the common logarithm (base 10) and the natural logarithm (base $e$). Using logarithms allows us to bring the variable exponent down, turning the exponential equation into a more manageable algebraic equation.

To effectively solve exponential equations, it's vital to grasp the fundamental principles of exponents and logarithms. Exponents indicate the number of times a base is multiplied by itself, while logarithms reveal the power to which a base must be raised to obtain a specific number. The relationship between these two concepts is inverse, similar to the relationship between addition and subtraction, or multiplication and division. This inverse connection is the cornerstone of solving exponential equations using logarithms. By applying logarithms to both sides of an exponential equation, we can effectively "undo" the exponentiation, transforming the problem into a simpler, linear form that is easier to solve for the unknown variable.

Moreover, it's crucial to remember the properties of logarithms, such as the product rule, quotient rule, and power rule. These rules enable us to manipulate logarithmic expressions and further simplify equations. For instance, the power rule, which states that $\log_b(x^p) = p \log_b(x)$, is particularly useful when dealing with exponents in logarithmic expressions. A solid understanding of these logarithmic properties is essential for successfully navigating the process of solving exponential equations. This foundational knowledge equips us to convert exponential equations into a form where variables can be easily isolated and solved, unlocking the solutions to a wide array of mathematical problems.

Step-by-Step Solution of $4^{x+3} = 7^{x-6}$

Let's tackle the equation $4^{x+3} = 7^{x-6}$ step by step. The first step involves applying a logarithm to both sides of the equation. The choice of logarithm base is arbitrary, but the common logarithm (base 10) or the natural logarithm (base $e$) are frequently used for convenience. In this case, we'll use the natural logarithm (ln). This gives us:

$\ln(4^{x+3}) = \ln(7^{x-6})$

Next, we apply the power rule of logarithms, which states that $\ln(a^b) = b \ln(a)$. This allows us to bring the exponents down:

$(x+3)\ln(4) = (x-6)\ln(7)$

Now, we have a linear equation in terms of $x$. The next step is to distribute the logarithms:

$x\ln(4) + 3\ln(4) = x\ln(7) - 6\ln(7)$

Our goal is to isolate $x$, so we rearrange the equation to group the terms containing $x$ on one side and the constant terms on the other:

$x\ln(4) - x\ln(7) = -6\ln(7) - 3\ln(4)$

Factor out $x$ from the left side:

$x(\ln(4) - \ln(7)) = -6\ln(7) - 3\ln(4)$

Finally, divide both sides by $(\ln(4) - \ln(7))$ to solve for $x$:

$x = \frac{-6\ln(7) - 3\ln(4)}{\ln(4) - \ln(7)}$

This is the exact solution for $x$. To obtain a numerical approximation, we use a calculator to evaluate the logarithms and perform the division.

Numerical Approximation and Final Answer

To find the numerical approximation of the solution, we need to calculate the values of the natural logarithms and perform the arithmetic operations. Using a calculator, we find that:

• $\ln(4) \approx 1.386294$
• $\ln(7) \approx 1.945910$

Substituting these values into the equation for $x$, we get:

$x \approx \frac{-6(1.945910) - 3(1.386294)}{1.386294 - 1.945910}$

$x \approx \frac{-11.67546 - 4.158882}{-0.559616}$

$x \approx \frac{-15.834342}{-0.559616}$

$x \approx 28.294$

Rounding the final answer to three decimal places, as requested, we have:

$x \approx 28.294$

Therefore, the solution to the exponential equation $4^{x+3} = 7^{x-6}$, rounded to three decimal places, is $x \approx 28.294$. This process demonstrates how logarithms can be effectively used to solve exponential equations by transforming them into linear equations that are easier to manipulate and solve.

Alternative Methods and Considerations

While using logarithms is the most common method for solving exponential equations, there are alternative approaches and considerations depending on the specific equation. For example, if the bases can be expressed as powers of the same number, the equation can sometimes be simplified without logarithms. Consider the equation $2^{2x} = 4^{x-1}$. Since $4 = 2^2$, we can rewrite the equation as $2^{2x} = (2^2)^{x-1}$, which simplifies to $2^{2x} = 2^{2x-2}$. In this case, we can equate the exponents directly: $2x = 2x - 2$. However, this leads to a contradiction ($0 = -2$), indicating that there is no solution to this particular equation.

Another consideration is the choice of logarithm base. While the natural logarithm (base $e$) and the common logarithm (base 10) are frequently used, any base can be used as long as it is applied consistently to both sides of the equation. The change of base formula, $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, can be useful if you need to convert between different bases.

Furthermore, it's important to check for extraneous solutions, especially when dealing with more complex equations. Extraneous solutions are values that satisfy the transformed equation but not the original equation. This can occur when applying logarithmic properties, as logarithms are only defined for positive arguments. For instance, if solving an equation involves taking the logarithm of an expression containing $x$, the solution must be checked to ensure that the expression remains positive.

In summary, while logarithms provide a powerful tool for solving exponential equations, it's crucial to understand the underlying principles, consider alternative methods when appropriate, and always verify the solutions to avoid extraneous results. These considerations ensure the accuracy and validity of the solutions obtained.

Conclusion

In conclusion, using logarithms is a powerful technique for solving exponential equations. By applying logarithms to both sides of the equation and utilizing logarithmic properties, we can transform exponential equations into linear equations that are easier to solve. We demonstrated this method with the example equation $4^{x+3} = 7^{x-6}$, walking through each step, from applying the logarithm to isolating the variable and finding a numerical approximation. Remember to round the final answer to the specified decimal places and to check your solutions.

Understanding the principles behind exponential equations and logarithms is crucial for solving a wide range of mathematical problems in various fields. Whether dealing with exponential growth, decay, or other complex scenarios, the ability to manipulate and solve these equations is an invaluable skill. The step-by-step approach outlined in this article provides a solid foundation for tackling such problems, empowering you to confidently apply these techniques in different contexts. Practice is key to mastering these skills, so working through various examples will further solidify your understanding and proficiency in solving exponential equations using logarithms.

This article has provided a comprehensive guide to solving exponential equations using logarithms, covering the essential steps, alternative methods, and important considerations. With this knowledge, you can confidently approach and solve exponential equations, expanding your mathematical toolkit and enhancing your problem-solving capabilities.