Solving Exponential Equations A Step-by-Step Guide For 2^(-x-5) = 9^(10x)

In this article, we will delve into the process of solving for x in the equation 2^(-x-5) = 9^(10x). This equation involves exponential terms, and to find the exact value of x, we will employ logarithms. Logarithms are a powerful tool for solving equations where the variable is in the exponent. We will explore the steps necessary to isolate x and express the solution using both base-10 and base-e logarithms (natural logarithms).

Understanding Exponential Equations

To effectively solve exponential equations, a solid grasp of exponential functions and their properties is essential. An exponential function is a function of the form f(x) = a^x, where a is a constant base and x is the exponent. Key properties of exponential functions include the fact that exponential functions are defined for all real numbers and that they grow rapidly as x increases, especially when the base a is greater than 1. When solving exponential equations, the goal is to isolate the variable exponent. This often involves using logarithms to "undo" the exponential function. Logarithms and exponential functions are inverse operations, meaning that they effectively cancel each other out. For instance, if we have an equation like a^x = b, we can take the logarithm of both sides with base a to get x = log_a(b).

Logarithms as a Tool

Logarithms are the inverse operation to exponentiation. The logarithm of a number b to the base a is the exponent to which a must be raised to produce b. Mathematically, this is written as logₐ(b) = x, which means aˣ = b. There are two commonly used logarithms: the base-10 logarithm (denoted as log₁₀ or simply log) and the natural logarithm (base e, denoted as ln). Both base-10 and natural logarithms are instrumental in solving exponential equations. The choice between them often depends on the context and the form of the equation. Base-10 logarithms are useful when dealing with powers of 10, while natural logarithms are particularly useful in calculus and other advanced mathematical contexts. The key property of logarithms that we will use in solving the given equation is the power rule, which states that logₐ(mⁿ) = n logₐ(m). This rule allows us to bring the exponent down as a coefficient, which is crucial for isolating the variable x.

Step-by-Step Solution

Let's walk through the process of solving the equation 2^(-x-5) = 9^(10x) step by step. We will use logarithms to bring the exponents down and isolate x. The beauty of using logarithms lies in their ability to simplify exponential equations, making them easier to solve algebraically. By applying the properties of logarithms, we can transform what initially seems like a complex equation into a more manageable linear equation.

1. Apply Logarithms to Both Sides

To begin, we apply a logarithm to both sides of the equation. We can use either the base-10 logarithm (log) or the natural logarithm (ln). For this solution, let's use the natural logarithm (ln):

ln(2^(-x-5)) = ln(9^(10x))

Applying the logarithm to both sides allows us to use the power rule of logarithms, which will help in simplifying the equation and isolating the variable x. This is a fundamental step in solving exponential equations, as it transforms the problem into a more solvable form.

2. Use 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 gives:

(-x - 5) * ln(2) = 10x * ln(9)

This step is crucial because it moves the exponents from being superscripts to being coefficients, which allows us to treat the equation as a linear equation in terms of x. The power rule is one of the most important properties of logarithms for solving exponential equations, and it is used extensively in various mathematical and scientific fields.

3. Expand and Rearrange the Equation

Next, we expand the left side of the equation and rearrange the terms to group the x terms together:

-x * ln(2) - 5 * ln(2) = 10x * ln(9)

Now, we want to isolate x. To do this, we move all terms containing x to one side of the equation and the constant terms to the other side:

-5 * ln(2) = 10x * ln(9) + x * ln(2)

This rearrangement sets up the equation for factoring out x, which is the next key step in solving for the variable. Grouping the x terms allows us to combine them and eventually isolate x on one side of the equation.

4. Factor Out x

Factor out x from the right side of the equation:

-5 * ln(2) = x * (10 * ln(9) + ln(2))

Factoring out x is a crucial step because it allows us to express the equation in the form of a simple product, where one factor is x and the other is a constant. This form makes it straightforward to isolate x by dividing both sides of the equation by the constant factor.

5. Isolate x

To isolate x, divide both sides of the equation by (10 * ln(9) + ln(2)):

x = (-5 * ln(2)) / (10 * ln(9) + ln(2))

This step directly solves for x by undoing the multiplication in the previous step. The expression now gives the exact value of x in terms of natural logarithms. While this is a precise answer, it can be further simplified or expressed using base-10 logarithms if desired.

6. Exact Answer Using Base-e Logarithms

The exact answer using base-e logarithms (natural logarithms) is:

x = (-5 * ln(2)) / (10 * ln(9) + ln(2))

This expression provides the solution in its most accurate form, using natural logarithms. The value of x can be approximated using a calculator, but the exact form is important for theoretical purposes and precise calculations.

7. Exact Answer Using Base-10 Logarithms

To express the answer using base-10 logarithms, we use the change of base formula, which states that ln(a) = log(a) / log(e). Thus, we can rewrite the solution in terms of base-10 logarithms:

x = (-5 * log(2) / log(e)) / (10 * log(9) / log(e) + log(2) / log(e))

We can simplify this by multiplying the numerator and denominator by log(e), which cancels out the log(e) terms:

x = (-5 * log(2)) / (10 * log(9) + log(2))

This is the exact answer using base-10 logarithms. Similar to the natural logarithm form, this expression provides a precise value for x and can be used for accurate calculations.

Alternative Approaches and Insights

While using logarithms is the standard method for solving exponential equations, understanding alternative approaches and gaining insights into the properties of exponential and logarithmic functions can further enhance your problem-solving skills. One alternative approach is to use numerical methods to approximate the solution. Numerical methods, such as the Newton-Raphson method, can provide highly accurate approximations of the roots of equations, especially when analytical solutions are difficult to obtain. However, for the purpose of finding an exact answer, logarithms are the most effective tool.

Insights into Exponential and Logarithmic Functions

Exponential functions and logarithms are fundamental in mathematics and have wide-ranging applications in various fields, including physics, engineering, and economics. Understanding their properties and behavior is crucial for solving complex problems and modeling real-world phenomena. Exponential functions, such as f(x) = a^x, represent situations where the rate of change is proportional to the current value. This property makes them ideal for modeling growth and decay processes, such as population growth, radioactive decay, and compound interest. Logarithmic functions, on the other hand, are used to "undo" exponential functions and are particularly useful for solving equations where the variable is in the exponent. They also have applications in measuring the magnitude of earthquakes (Richter scale), sound intensity (decibels), and acidity (pH scale).

Common Mistakes to Avoid

When solving exponential equations, it's easy to make mistakes if you're not careful. Here are some common errors to watch out for:

1. Incorrectly Applying Logarithm Properties: The power rule, product rule, and quotient rule of logarithms must be applied correctly. A common mistake is to misapply the power rule or to attempt to apply logarithm rules to sums or differences that only apply to products and quotients.
2. Algebraic Errors: Mistakes in algebraic manipulation, such as incorrectly distributing terms or combining like terms, can lead to incorrect solutions. It's important to carefully check each step and ensure that the algebraic manipulations are correct.
3. Forgetting the Base of the Logarithm: When applying logarithms, it's crucial to maintain consistency in the base. If using natural logarithms (base e), ensure all logarithms are natural logarithms. Similarly, if using base-10 logarithms, maintain the base throughout the solution.

By being aware of these common pitfalls, you can avoid errors and ensure accurate solutions.

Conclusion

In this article, we successfully solved for x in the equation 2^(-x-5) = 9^(10x) using logarithms. We found the exact answers using both base-e (natural) and base-10 logarithms. The key steps involved applying logarithms to both sides, using the power rule to bring down exponents, rearranging the equation to isolate x, and expressing the solution in terms of logarithms. Understanding exponential and logarithmic functions is essential for solving such equations, and the techniques discussed here can be applied to a wide range of problems in mathematics and related fields. By mastering these methods, you can confidently tackle complex exponential equations and gain a deeper understanding of the mathematical principles involved.

The final answer is:

$$x = \frac{-5 \ln(2)}{10 \ln(9) + \ln(2)}$$

Or, using base-10 logarithms:

$$x = \frac{-5 \log(2)}{10 \log(9) + \log(2)}$$