Solving 3^x = 62 A Step-by-Step Guide

When faced with an exponential equation like 3^x = 62, finding the value of 'x' might seem daunting at first. However, with the right techniques, these equations become quite manageable. This article will guide you through the process of solving such equations, focusing on the power of logarithms. We'll break down the steps, explain the underlying concepts, and provide a clear, concise solution to the equation 3^x = 62, rounded to the nearest hundredth. By the end of this guide, you'll not only be able to solve this specific problem but also gain a solid understanding of how to approach similar exponential equations.

Understanding Exponential Equations

Before diving into the solution, it's crucial to understand what exponential equations are and why they require a different approach than typical algebraic equations. In an exponential equation, the variable appears in the exponent, as seen in 3^x = 62. This means 'x' represents the power to which the base (3 in this case) must be raised to obtain the result (62).

The key challenge here is that we can't simply isolate 'x' using standard algebraic operations like addition, subtraction, multiplication, or division. To solve for an exponent, we need a tool that can "undo" the exponentiation, and that tool is the logarithm. Logarithms are essentially the inverse operation of exponentiation. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Understanding this inverse relationship is fundamental to solving exponential equations. For instance, if we have b^y = x, the logarithmic form is log_b(x) = y, where 'b' is the base, 'y' is the exponent, and 'x' is the result. Grasping this concept is the first step in mastering exponential equation solutions. We often encounter exponential equations in various fields, from finance (compound interest) to science (radioactive decay), making the ability to solve them a valuable skill.

The Power of Logarithms

Logarithms are the cornerstone of solving exponential equations. They provide a way to "bring down" the exponent, making it a coefficient that can be manipulated using standard algebraic techniques. The fundamental property we'll use is the power rule of logarithms, which states that log_b(a^c) = c * log_b(a). This rule is the key to unlocking the value of 'x' in our equation.

The beauty of logarithms lies in their ability to transform an exponential equation into a linear one, which is far easier to solve. When we apply a logarithm to both sides of an exponential equation, the power rule allows us to move the exponent from its elevated position to a more accessible one as a multiplier. This transformation is crucial because it allows us to isolate the variable 'x' and solve for it. While various bases can be used for logarithms, the two most common are the base-10 logarithm (log) and the natural logarithm (ln, base e). The choice of base doesn't affect the final answer, but it's important to use a base that's readily available on your calculator. For example, the equation 3^x = 62 can be transformed using logarithms to isolate 'x'. By applying the power rule, we essentially convert a difficult problem into a manageable one. This makes logarithms an indispensable tool in mathematics, science, and engineering, especially when dealing with growth, decay, and other phenomena that can be modeled exponentially. Therefore, understanding how to apply logarithms is not just about solving equations; it's about gaining a deeper understanding of the mathematical relationships that govern the world around us.

Solving 3^x = 62 Using Logarithms

Now, let's apply the power of logarithms to solve our equation, 3^x = 62. The first step is to take the logarithm of both sides of the equation. We can use either the common logarithm (base 10) or the natural logarithm (base e). For this example, let's use the common logarithm (log), but the process is the same with the natural logarithm (ln). Applying the logarithm to both sides gives us log(3^x) = log(62).

Next, we use the power rule of logarithms, which states that log_b(a^c) = c * log_b(a). Applying this rule to our equation, we get x * log(3) = log(62). Notice how the exponent 'x' has now become a coefficient, making it possible to isolate it. To isolate 'x', we simply divide both sides of the equation by log(3), resulting in x = log(62) / log(3). Now, we can use a calculator to find the decimal values of log(62) and log(3). We find that log(62) ≈ 1.7924 and log(3) ≈ 0.4771. Substituting these values into our equation, we get x ≈ 1.7924 / 0.4771. Performing the division, we find x ≈ 3.7565. Finally, we round the answer to the nearest hundredth, as requested, giving us x ≈ 3.76. This step-by-step approach demonstrates how logarithms effectively transform an exponential equation into a simple algebraic one. Understanding each step is key to confidently solving similar problems in the future. By mastering this technique, you'll be equipped to tackle a wide range of exponential equations that arise in various mathematical and scientific contexts.

Verification and Alternative Approaches

To ensure our solution x ≈ 3.76 is correct, it's always a good practice to verify it. We can do this by substituting the value back into the original equation: 3^3.76. Using a calculator, we find that 3^3.76 ≈ 61.98, which is very close to 62. The slight difference is due to rounding errors, but it confirms that our solution is accurate to the nearest hundredth.

Besides using the common logarithm (base 10), we could have also used the natural logarithm (ln) to solve the equation. The process is exactly the same, but we would have started with ln(3^x) = ln(62). Applying the power rule, we get x * ln(3) = ln(62). Then, we isolate 'x' by dividing both sides by ln(3), resulting in x = ln(62) / ln(3). Using a calculator, we find that ln(62) ≈ 4.1271 and ln(3) ≈ 1.0986. Substituting these values, we get x ≈ 4.1271 / 1.0986, which gives us x ≈ 3.7565, the same result we obtained using common logarithms. This demonstrates that the choice of logarithm base does not affect the final solution. The important thing is to consistently apply the power rule and perform the calculations accurately. This verification step highlights the importance of understanding the properties of logarithms and their versatility in solving exponential equations. It also underscores the value of checking your work to ensure accuracy, especially when dealing with approximations and rounding.

Conclusion

In conclusion, we have successfully solved the exponential equation 3^x = 62 by leveraging the power of logarithms. We walked through the steps of taking the logarithm of both sides, applying the power rule, isolating 'x', and calculating the result. We found that x ≈ 3.76 when rounded to the nearest hundredth. We also verified our solution by substituting it back into the original equation and explored an alternative approach using the natural logarithm, further solidifying our understanding of the process. The ability to solve exponential equations is a valuable skill in mathematics and various scientific disciplines. By mastering the techniques outlined in this guide, you'll be well-equipped to tackle a wide range of similar problems. Remember, the key to success lies in understanding the fundamental properties of logarithms and applying them systematically. Practice is also crucial. The more you work with exponential equations and logarithms, the more comfortable and confident you'll become in solving them. So, continue to explore, experiment, and apply these concepts to new and challenging problems. With dedication and practice, you'll unlock the full potential of logarithms and their applications.

Final Answer: B) x=3.76

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