Solving $18^{10x} = 13^{x+3}$ A Step By Step Guide With Logarithms

Exponential equations, where the variable appears in the exponent, can seem daunting at first. However, with a systematic approach and a solid understanding of logarithmic properties, these equations can be solved elegantly. This article delves into the solution of the exponential equation $18^{10x} = 13^{x+3}$, providing a step-by-step guide and highlighting the underlying mathematical principles. Exponential equations are a fundamental topic in algebra and calculus, appearing in various applications such as modeling population growth, radioactive decay, and compound interest. Mastering the techniques for solving these equations is crucial for anyone pursuing studies in mathematics, science, or engineering. In this comprehensive guide, we will explore the intricacies of solving the equation, ensuring a clear and thorough understanding for readers of all backgrounds. Our approach will emphasize clarity, providing detailed explanations for each step and highlighting key concepts. By the end of this article, you will be equipped with the knowledge and skills to tackle similar exponential equations with confidence.

Understanding the Problem

Before diving into the solution, let's first understand the given equation: $18^{10x} = 13^{x+3}$. We are tasked with finding the value of x that satisfies this equation. The presence of x in the exponents of both sides necessitates the use of logarithms. Logarithms are the inverse operation to exponentiation, allowing us to "bring down" the variable from the exponent. The key to solving this equation lies in applying the properties of logarithms correctly. We will explore these properties in detail as we proceed. Recognizing the structure of the equation is the first step towards finding the solution. The equation involves two exponential terms with different bases (18 and 13) and exponents that are linear expressions in x. This form is typical of exponential equations encountered in various mathematical contexts. By understanding the problem's structure, we can choose the appropriate tools and techniques to solve it effectively. The next step is to apply logarithms to both sides of the equation, a crucial step that transforms the exponential equation into a more manageable algebraic form. This transformation is based on the fundamental property of logarithms that allows us to move exponents to the front as multipliers.

Applying Logarithms

The core strategy for solving exponential equations is to apply logarithms to both sides. This is because the logarithm function has the property that $\log(a^b) = b \log(a)$, which allows us to bring the exponents down as coefficients. We can use either base-10 logarithms (denoted as log) or natural logarithms (base-e, denoted as ln). For this example, let's use the natural logarithm (ln). Applying the natural logarithm to both sides of the equation $18^{10x} = 13^{x+3}$, we get:

$\ln(18^{10x}) = \ln(13^{x+3})$

Now, using the logarithmic property mentioned above, we can rewrite the equation as:

$10x \ln(18) = (x+3) \ln(13)$

This step is critical as it transforms the exponential equation into a linear equation in x, which is much easier to solve. The application of logarithms simplifies the problem significantly by eliminating the exponential terms. The choice of using natural logarithms is arbitrary; base-10 logarithms would work equally well. The key is to apply the same logarithm to both sides of the equation to maintain equality. This transformation is a fundamental technique in solving exponential equations, and it is applicable to a wide range of similar problems. The resulting equation, $10x \ln(18) = (x+3) \ln(13)$, is a linear equation in x, albeit with logarithmic coefficients. The next step involves expanding the equation and collecting terms to isolate x.

Expanding and Rearranging

Next, we expand the right side of the equation and rearrange the terms to isolate x. Starting from:

$10x \ln(18) = (x+3) \ln(13)$

We distribute $\ln(13)$ on the right side:

$10x \ln(18) = x \ln(13) + 3 \ln(13)$

Now, we want to group the terms containing x on one side of the equation. Subtract $x \ln(13)$ from both sides:

$10x \ln(18) - x \ln(13) = 3 \ln(13)$

Factor out x from the left side:

$x(10 \ln(18) - \ln(13)) = 3 \ln(13)$

This step of expanding and rearranging is crucial for isolating the variable x. By distributing the logarithm and collecting like terms, we transform the equation into a form where x can be easily solved for. The factoring step is particularly important as it allows us to express the equation as a product of x and a constant, making the final step of division straightforward. The expression $10 \ln(18) - \ln(13)$ represents a constant value, which can be computed using a calculator if a numerical approximation is desired. However, for an exact answer, we will leave it in this form. The next step is to divide both sides of the equation by this constant to solve for x.

Solving for x

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

$x = \frac{3 \ln(13)}{10 \ln(18) - \ln(13)}$

This is the exact solution for x using natural logarithms. We can also express the solution using base-10 logarithms. To do this, we simply replace \ln with log:

$x = \frac{3 \log(13)}{10 \log(18) - \log(13)}$

Both expressions are equivalent and represent the exact value of x. This step concludes the algebraic solution of the equation. We have successfully isolated x and expressed it in terms of logarithms. The solution represents the precise value of x that satisfies the original exponential equation. While a numerical approximation can be obtained using a calculator, the exact solution provides a more complete and accurate representation. The expression for x involves logarithmic terms, reflecting the fundamental role of logarithms in solving exponential equations. The solution highlights the importance of understanding and applying logarithmic properties correctly. The next step, if desired, would be to approximate the value of x using a calculator.

Approximating the Solution (Optional)

If we need a numerical approximation, we can use a calculator to evaluate the expression. Using a calculator, we find that:

$\ln(13) \approx 2.5649$

$\ln(18) \approx 2.8904$

Substituting these values into the expression for x:

$x \approx \frac{3 \times 2.5649}{10 \times 2.8904 - 2.5649}$

$x \approx \frac{7.6947}{28.904 - 2.5649}$

$x \approx \frac{7.6947}{26.3391}$

$x \approx 0.2921$

So, the approximate value of x is 0.2921. This step demonstrates how to obtain a numerical approximation of the solution. While the exact solution provides a precise representation, the approximation gives a practical value that can be used in applications. The approximation is obtained by substituting the numerical values of the logarithmic terms into the expression for x and performing the arithmetic operations. The accuracy of the approximation depends on the precision of the logarithmic values used. For most practical purposes, a four-decimal-place approximation is sufficient. The approximate solution provides a concrete value for x, allowing us to verify the solution and interpret its significance in the context of the original problem.

Verification

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

$18^{10 \times 0.2921} \approx 13^{0.2921 + 3}$

$18^{2.921} \approx 13^{3.2921}$

Using a calculator:

$18^{2.921} \approx 832.2$

$13^{3.2921} \approx 832.2$

Since both sides are approximately equal, our solution is verified. This step is essential to ensure the correctness of the solution. By substituting the approximate value of x back into the original equation, we can check if both sides are equal. The verification process confirms that our algebraic manipulations and calculations were accurate. The slight difference in the values is due to rounding errors in the approximation. If a more precise approximation is used, the equality will be more accurate. The verification step provides confidence in the solution and ensures that it satisfies the original equation. It is a crucial part of the problem-solving process, especially in mathematics.

Conclusion

In this article, we have demonstrated a step-by-step solution for the exponential equation $18^{10x} = 13^{x+3}$. We applied logarithms to both sides, used logarithmic properties to simplify the equation, rearranged terms to isolate x, and obtained the exact solution:

$x = \frac{3 \ln(13)}{10 \ln(18) - \ln(13)}$ or $x = \frac{3 \log(13)}{10 \log(18) - \log(13)}$

We also found an approximate solution, $x \approx 0.2921$, and verified our result. Solving exponential equations requires a good understanding of logarithms and their properties. By following a systematic approach, these equations can be solved with confidence. This article has provided a comprehensive guide to solving one such equation, highlighting the key steps and principles involved. The process of solving exponential equations is a fundamental skill in mathematics and has applications in various fields, including science, engineering, and finance. Mastering this skill is essential for anyone pursuing further studies in these areas. The techniques discussed in this article can be applied to a wide range of exponential equations, making it a valuable resource for students and professionals alike. The conclusion summarizes the key steps and results of the solution, reinforcing the understanding of the process.