Solving Exponential Equations With Logarithms

JU07/16/2025 08, 2025 by THE IDEN 46 views

This article delves into the methods for solving exponential equations, focusing on leveraging logarithms to isolate the variable. We will explore two distinct examples, providing a step-by-step breakdown of the solution process for each. These examples will illustrate how to effectively utilize logarithmic properties to simplify equations and arrive at accurate solutions. We aim to equip you with the necessary tools and understanding to confidently tackle a variety of exponential equations.

Problem 1: Solving 5e^(-3x) = 16

To solve the equation 5e^(-3x) = 16, we aim to isolate x. This requires a series of algebraic manipulations, focusing on undoing the operations applied to x. Our primary tool in this process will be the natural logarithm, which is the logarithm to the base e. The natural logarithm possesses a crucial property: ln(e^x) = x, which allows us to effectively "undo" the exponential function with base e. Mastering the application of logarithms is crucial for solving exponential equations, making it an indispensable technique in various scientific and engineering fields. Exponential equations themselves are vital for modeling real-world phenomena, including population growth, radioactive decay, and compound interest. Therefore, the ability to solve them accurately is essential for making predictions and informed decisions in diverse contexts. The following steps outline the solution:

Isolate the Exponential Term: Begin by isolating the exponential term, e^(-3x). This is achieved by dividing both sides of the equation by 5:
```
5e^(-3x) = 16
e^(-3x) = 16/5
```
Isolating the exponential term is a crucial first step because it allows us to directly apply the logarithm. Without this step, we would be dealing with a more complex expression, and applying the logarithm would not be as straightforward. This initial isolation simplifies the subsequent steps and makes the equation more manageable.
Apply the Natural Logarithm: Apply the natural logarithm (ln) to both sides of the equation. This step utilizes the property that ln(e^x) = x to eliminate the exponential function on the left side:
```
ln(e^(-3x)) = ln(16/5)
-3x = ln(16/5)
```
The application of the natural logarithm is the key to solving the equation. By taking the natural logarithm of both sides, we effectively bring the exponent down as a coefficient, transforming the exponential equation into a linear one. This is a fundamental technique in solving exponential equations and is based on the inverse relationship between exponential and logarithmic functions.
Solve for x: Divide both sides by -3 to isolate x:
```
x = ln(16/5) / -3
x ≈ -0.3916
```
This final step involves a simple division to isolate x. The result is the solution to the original equation. We can use a calculator to approximate the value of ln(16/5) and then divide by -3 to obtain the numerical solution. The approximate solution, -0.3916, provides a concrete value for x that satisfies the equation. This step highlights the importance of basic algebraic operations in solving mathematical problems.

Therefore, the solution to the equation 5e^(-3x) = 16 is approximately x ≈ -0.3916. This process demonstrates the power of logarithms in solving exponential equations, providing a systematic approach to isolate the variable and find the solution.

Problem 2: Solving 3^x * 4^(x+2) = 8

This problem requires a slightly different approach due to the presence of multiple exponential terms with different bases. The key here is to again utilize logarithms, but this time we'll focus on leveraging the properties of logarithms to combine terms. Specifically, we will use the property log(a * b) = log(a) + log(b) and the power rule log(a^b) = b * log(a). These properties allow us to break down complex exponential expressions into simpler, more manageable terms. This approach is crucial for solving equations where variables appear in multiple exponents or with different bases. Exponential equations of this type frequently arise in modeling complex systems, such as those found in finance and physics. The following steps illustrate the solution process:

Apply the Logarithm: Take the logarithm of both sides. We can use any base logarithm, but the common logarithm (base 10) is often convenient. Applying the logarithm to both sides allows us to utilize logarithmic properties to simplify the equation:
```
log(3^x * 4^(x+2)) = log(8)
```
Applying the logarithm is a crucial step because it transforms the exponential equation into a form where we can use logarithmic properties to simplify it. This step sets the stage for breaking down the complex expression into more manageable parts. The choice of logarithm base is arbitrary, but using a common base like 10 often simplifies calculations.
Use Logarithmic Properties to Expand: Use the logarithmic property log(a * b) = log(a) + log(b) to expand the left side of the equation:
```
log(3^x) + log(4^(x+2)) = log(8)
```
This expansion is a key application of logarithmic properties. By breaking the product inside the logarithm into a sum of logarithms, we begin to separate the terms containing the variable x. This separation is essential for isolating x and solving the equation. Understanding and applying logarithmic properties is crucial for effectively manipulating logarithmic expressions.
Apply the Power Rule: Apply the power rule log(a^b) = b * log(a) to further expand the terms:
```
x * log(3) + (x+2) * log(4) = log(8)
```
The power rule is another fundamental logarithmic property that is crucial for solving exponential equations. By bringing the exponents down as coefficients, we transform the exponential terms into linear terms, making the equation easier to solve. This step is a significant simplification and moves us closer to isolating x.
Distribute and Rearrange: Distribute log(4) and rearrange the equation to group terms with x:
```
x * log(3) + x * log(4) + 2 * log(4) = log(8)
x * log(3) + x * log(4) = log(8) - 2 * log(4)
```
Distributing and rearranging the terms is a standard algebraic technique used to group like terms. In this case, we group the terms containing x on one side of the equation and the constant terms on the other. This rearrangement is essential for factoring out x and isolating it.
Factor out x: Factor out x from the terms on the left side:
```
x * (log(3) + log(4)) = log(8) - 2 * log(4)
```
Factoring out x is a key step in isolating the variable. By factoring, we combine the terms containing x into a single term, making it possible to divide and solve for x. This is a common technique used in solving algebraic equations.
Solve for x: Divide both sides by (log(3) + log(4)) to isolate x:
```
x = (log(8) - 2 * log(4)) / (log(3) + log(4))
x ≈ -0.4259
```
This final step involves dividing both sides of the equation by the coefficient of x to isolate the variable. The result is the solution to the original equation. We can use a calculator to approximate the values of the logarithms and then perform the calculations to obtain the numerical solution. The approximate solution, -0.4259, provides a concrete value for x that satisfies the equation.

Therefore, the solution to the equation 3^x * 4^(x+2) = 8 is approximately x ≈ -0.4259. This example highlights the importance of logarithmic properties in simplifying complex exponential equations and demonstrates a systematic approach to finding the solution.

Conclusion

In summary, solving exponential equations often involves the strategic application of logarithms. Whether it's isolating a single exponential term or dealing with multiple terms with different bases, understanding and utilizing logarithmic properties is crucial. The examples presented demonstrate the step-by-step processes involved, from applying the natural logarithm to utilizing logarithmic properties like the product rule and power rule. Mastering these techniques empowers you to tackle a wide range of exponential equations and their applications in various fields. Exponential equations are fundamental in modeling numerous real-world phenomena, making their solutions critical for analysis and prediction. Therefore, a solid understanding of these methods is an invaluable asset in mathematics, science, and engineering.