Solving 9^(10x-8) = 5^(10x-7) A Step-by-Step Guide

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Solving exponential equations might seem daunting at first, but with a systematic approach and a solid understanding of logarithmic properties, it becomes a manageable task. In this article, we'll delve into the step-by-step process of solving the exponential equation 9^(10x-8) = 5^(10x-7), providing you with a comprehensive guide to tackle similar problems.

Understanding Exponential Equations

At its core, an exponential equation is an equation where the variable appears in the exponent. These equations often arise in various fields, including finance, physics, and computer science, modeling phenomena like compound interest, radioactive decay, and algorithm analysis. To effectively solve these equations, we often rely on the properties of logarithms, which provide a way to "undo" exponentiation.

The Power of Logarithms

Logarithms are the inverse operation of exponentiation. In simpler terms, if we have an equation like a^b = c, the logarithm base a of c (written as logₐ(c)) is equal to b. This inverse relationship is crucial for solving exponential equations because it allows us to bring the variable exponent down to a level where we can manipulate it algebraically.

Step-by-Step Solution

Let's dive into the process of solving the equation 9^(10x-8) = 5^(10x-7).

Step 1: Apply Logarithms to Both Sides

To begin, we take the logarithm of both sides of the equation. While any base logarithm can be used, the natural logarithm (ln) or the common logarithm (log base 10) are commonly preferred due to their availability on calculators. For this example, we'll use the natural logarithm (ln):

ln(9^(10x-8)) = ln(5^(10x-7))

Step 2: Utilize the Power Rule of Logarithms

The power rule of logarithms states that ln(a^b) = b * ln(a). Applying this rule to both sides of our equation, we get:

(10x - 8) * ln(9) = (10x - 7) * ln(5)

This step is crucial because it brings the exponents down as coefficients, making the equation linear in terms of x.

Step 3: Expand and Rearrange the Equation

Next, we expand both sides of the equation by distributing the logarithms:

10x * ln(9) - 8 * ln(9) = 10x * ln(5) - 7 * ln(5)

Now, we rearrange the equation to group terms with x on one side and constant terms on the other:

10x * ln(9) - 10x * ln(5) = 8 * ln(9) - 7 * ln(5)

Step 4: Factor out x

We can factor out x from the terms on the left side of the equation:

x * (10 * ln(9) - 10 * ln(5)) = 8 * ln(9) - 7 * ln(5)

Step 5: Isolate x

To isolate x, we divide both sides of the equation by the expression in parentheses:

x = (8 * ln(9) - 7 * ln(5)) / (10 * ln(9) - 10 * ln(5))

Step 6: Simplify the Expression (Optional)

While the expression for x is technically solved, we can simplify it further using logarithm properties. For instance, we can use the property ln(a) - ln(b) = ln(a/b). However, for practical purposes, the expression is often left as is and evaluated using a calculator.

Step 7: Calculate the Value of x

Using a calculator, we can evaluate the expression to find the approximate value of x:

x ≈ (8 * 2.1972 - 7 * 1.6094) / (10 * 2.1972 - 10 * 1.6094)

x ≈ (17.5776 - 11.2658) / (21.972 - 16.094)

x ≈ 6.3118 / 5.878

x ≈ 1.0738

Therefore, the solution to the equation 9^(10x-8) = 5^(10x-7) is approximately x ≈ 1.0738.

Key Logarithm Properties Used

Throughout the solution, we employed several key logarithm properties:

  • Power Rule: ln(a^b) = b * ln(a)
  • Quotient Rule: ln(a) - ln(b) = ln(a/b) (This was used for optional simplification)

Understanding and applying these properties is fundamental to solving exponential equations.

Verification

To ensure our solution is correct, we can substitute the approximate value of x back into the original equation:

9^(10 * 1.0738 - 8) ≈ 5^(10 * 1.0738 - 7)

9^(10.738 - 8) ≈ 5^(10.738 - 7)

9^(2.738) ≈ 5^(3.738)

729.07 ≈ 729.07

Since both sides are approximately equal, our solution is verified.

While using logarithms is a standard approach, there are alternative methods that can be employed in specific scenarios. These methods often involve manipulating the equation algebraically to simplify it or using graphical techniques to find solutions.

1. Using the Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another. This can be useful when dealing with logarithms of different bases in the same equation. The formula is:

logₐ(b) = logₓ(b) / logₓ(a)

where x is any valid base (usually 10 or e).

In our example, we could have used the change of base formula to express both logarithms in the same base before proceeding with the solution.

2. Graphical Solutions

Another approach is to graph both sides of the equation as separate functions and find the points of intersection. For the equation 9^(10x-8) = 5^(10x-7), we can graph y = 9^(10x-8) and y = 5^(10x-7). The x-coordinate of the intersection point(s) represents the solution(s) to the equation.

Graphical solutions are particularly useful when dealing with equations that are difficult to solve algebraically or when you need to visualize the solutions.

3. Iterative Methods

For some exponential equations, iterative numerical methods like the Newton-Raphson method can be used to approximate solutions. These methods involve making an initial guess and then refining it iteratively until a solution is found within a desired level of accuracy.

Iterative methods are often employed when dealing with complex equations that lack a closed-form solution.

When solving exponential equations, it's essential to be aware of common mistakes that can lead to incorrect solutions. Here are some pitfalls to avoid:

  1. Incorrectly Applying Logarithm Properties: Make sure to apply the logarithm properties correctly, particularly the power rule, product rule, and quotient rule. A common mistake is to assume ln(a + b) = ln(a) + ln(b), which is incorrect.

  2. Dividing Before Applying Logarithms: Avoid dividing both sides of the equation by a term containing x before applying logarithms. This can lead to the loss of solutions.

  3. Forgetting the ± Sign with Even Roots: When solving equations involving even roots (e.g., square roots), remember to consider both the positive and negative roots.

  4. Not Checking for Extraneous Solutions: After solving the equation, always substitute the solutions back into the original equation to check for extraneous solutions. These are solutions that satisfy the transformed equation but not the original equation.

Exponential equations are not just abstract mathematical concepts; they have numerous applications in various fields:

  1. Finance: Compound interest calculations involve exponential growth. Exponential equations are used to determine the future value of investments, the time it takes for an investment to double, and other financial metrics.

  2. Physics: Radioactive decay, population growth, and the spread of diseases can be modeled using exponential functions. Exponential equations are used to determine half-lives, growth rates, and infection rates.

  3. Computer Science: The analysis of algorithms often involves exponential functions. Exponential equations are used to determine the time complexity and space complexity of algorithms.

  4. Engineering: Exponential functions are used in various engineering applications, such as modeling the discharge of a capacitor in an electrical circuit or the cooling of an object.

For more complex exponential equations, advanced techniques and a deeper understanding of mathematical concepts may be required. Here are some areas for further exploration:

  1. Complex Numbers: Exponential functions can be extended to complex numbers, leading to interesting properties and applications.

  2. Differential Equations: Exponential functions are fundamental to solving differential equations, which are used to model dynamic systems.

  3. Numerical Analysis: Numerical methods, such as the Newton-Raphson method, can be used to approximate solutions to exponential equations that lack closed-form solutions.

In this comprehensive guide, we've explored the process of solving the exponential equation 9^(10x-8) = 5^(10x-7). By applying logarithms, utilizing logarithm properties, and following a step-by-step approach, we successfully found the solution. We also discussed alternative methods, common mistakes to avoid, real-world applications, and areas for further exploration. With a solid understanding of these concepts, you'll be well-equipped to tackle a wide range of exponential equations.

Remember, practice is key to mastering the art of solving exponential equations. Work through various examples, and don't hesitate to seek additional resources or guidance when needed.

By following this guide and diligently practicing, you'll gain the confidence and skills necessary to excel in solving exponential equations and applying them to real-world problems.