Solving For X In 3^x = 12 A Step By Step Guide

by Scholario Team 47 views

In this comprehensive guide, we will delve into the process of solving for x in the exponential equation 3^x = 12. This type of equation, where the variable appears in the exponent, requires a different approach than standard algebraic equations. We'll explore the underlying mathematical principles, the application of logarithms, and the step-by-step methods to arrive at the solution. Whether you're a student tackling algebra problems, a math enthusiast seeking to expand your knowledge, or someone looking to refresh your mathematical skills, this guide will provide you with a clear and detailed explanation.

Exponential equations are equations where the variable appears in the exponent. They are fundamental in various fields, including mathematics, physics, engineering, and finance. The basic form of an exponential equation is a^x = b, where a is the base, x is the exponent, and b is the result. To solve for x, we need to isolate it, which typically involves using logarithms. Logarithms are the inverse operation of exponentiation, meaning they "undo" the exponent. Specifically, the logarithm of a number b to the base a (written as logₐ(b)) is the exponent to which a must be raised to produce b. Understanding this relationship between exponents and logarithms is crucial for solving exponential equations. The equation 3^x = 12 falls into this category, where 3 is the base, x is the exponent we want to find, and 12 is the result. The challenge lies in determining what power we need to raise 3 to in order to get 12. This is where logarithms become invaluable.

Logarithms are essential for solving exponential equations. The logarithm function is the inverse of the exponential function. That is, if a^x = b, then logₐ(b) = x. This relationship allows us to isolate the variable x when it appears in the exponent. There are two commonly used logarithms: the common logarithm (log base 10) and the natural logarithm (log base e, where e is approximately 2.71828). The choice of which logarithm to use often depends on the context and the calculator or software available, but the natural logarithm is particularly useful in calculus and other advanced mathematical fields. In the context of solving 3^x = 12, we can take the logarithm of both sides of the equation. This step is valid because applying the same function to both sides of an equation preserves the equality. By using logarithms, we transform the exponential equation into a linear equation, making it easier to solve for x. The key is to choose a logarithm base that will simplify the equation. We can use either the common logarithm (log base 10) or the natural logarithm (ln), but for clarity and ease of computation, we will demonstrate both methods.

To solve 3^x = 12 using common logarithms (base 10), we follow these steps:

  1. Apply the Common Logarithm to Both Sides: Taking the common logarithm (log base 10) of both sides of the equation 3^x = 12 gives us log₁₀(3^x) = log₁₀(12). This step is crucial as it allows us to use the properties of logarithms to simplify the equation.

  2. Use the Power Rule of Logarithms: The power rule of logarithms states that logₐ(b^c) = c * logₐ(b). Applying this rule to the left side of the equation, we get x * log₁₀(3) = log₁₀(12). The power rule allows us to bring the exponent x down as a coefficient, effectively isolating it from the exponential term.

  3. Isolate x: To isolate x, we divide both sides of the equation by log₁₀(3): x = log₁₀(12) / log₁₀(3). This step is a straightforward algebraic manipulation to solve for x.

  4. Calculate the Logarithms: Using a calculator, we find the approximate values of the logarithms: log₁₀(12) ≈ 1.07918 and log₁₀(3) ≈ 0.47712. Accurate calculation of the logarithms is essential for obtaining the correct value of x.

  5. Solve for x: Divide the values: x ≈ 1.07918 / 0.47712 ≈ 2.262. Therefore, the solution for x is approximately 2.262. This is the value that, when used as the exponent for 3, will give a result of 12.

Alternatively, we can solve 3^x = 12 using natural logarithms (base e). Here’s the process:

  1. Apply the Natural Logarithm to Both Sides: Taking the natural logarithm (ln) of both sides of the equation 3^x = 12 gives us ln(3^x) = ln(12). Similar to using common logarithms, this step is crucial for applying the properties of logarithms.

  2. Use the Power Rule of Logarithms: The power rule of logarithms applies to natural logarithms as well: ln(b^c) = c * ln(b). Applying this rule to the left side, we get x * ln(3) = ln(12). Again, the power rule allows us to bring the exponent x down as a coefficient.

  3. Isolate x: To isolate x, we divide both sides of the equation by ln(3): x = ln(12) / ln(3). This step is a straightforward algebraic manipulation to solve for x.

  4. Calculate the Logarithms: Using a calculator, we find the approximate values of the natural logarithms: ln(12) ≈ 2.48491 and ln(3) ≈ 1.09861. Accurate calculation of the logarithms is essential for obtaining the correct value of x.

  5. Solve for x: Divide the values: x ≈ 2.48491 / 1.09861 ≈ 2.262. Thus, the solution for x is approximately 2.262, which is consistent with the solution obtained using common logarithms. This consistency confirms the correctness of the method and the solution.

To verify our solution, we can substitute x ≈ 2.262 back into the original equation, 3^x = 12, and check if it holds true:

3^2.262 ≈ 12

Using a calculator, we find that 3^2.262 ≈ 11.999, which is very close to 12. The slight difference is due to the rounding of the value of x. This verification step is crucial as it ensures that the solution we obtained is correct and that there were no errors in our calculations. It also reinforces the understanding of the relationship between exponential functions and logarithms.

When solving exponential equations, several common mistakes can occur. Being aware of these pitfalls can help prevent errors and ensure accurate solutions:

  1. Incorrectly Applying Logarithm Properties: One common mistake is misapplying the power rule or other logarithm properties. For example, logₐ(b + c) is not equal to logₐ(b) + logₐ(c). It's essential to remember and correctly apply the logarithm rules.

  2. Dividing Before Applying Logarithms: Another mistake is attempting to isolate the base before applying logarithms. For instance, in the equation 3^(x+1) = 12, one might try to divide by 3 first, which is incorrect. Logarithms should be applied before performing other algebraic manipulations on the exponent.

  3. Calculator Errors: Incorrectly entering values into a calculator or using the wrong function (e.g., confusing log and ln) can lead to errors. It’s important to double-check the calculator inputs and the function used.

  4. Rounding Errors: Rounding intermediate values too early can introduce significant errors in the final answer. It's best to keep as many decimal places as possible during the calculations and only round the final result.

  5. Forgetting the Definition of Logarithms: A fundamental misunderstanding of the relationship between logarithms and exponentials can lead to errors. Remember that logarithms are the inverse of exponential functions, and this relationship is key to solving these equations.

Exponential equations are not just theoretical mathematical constructs; they have numerous real-world applications. Understanding and solving these equations is crucial in various fields:

  1. Finance: Compound interest calculations rely heavily on exponential equations. The formula for compound interest, A = P(1 + r/n)^(nt), involves exponential terms, where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. Solving for different variables in this equation often requires using logarithms.

  2. Population Growth: Exponential growth models are used to describe population dynamics. The equation P(t) = P₀ * e^(kt) models the population P at time t, where P₀ is the initial population, k is the growth rate, and e is the base of the natural logarithm. Solving for time or growth rate involves solving exponential equations.

  3. Radioactive Decay: Radioactive decay follows an exponential pattern. The equation N(t) = N₀ * e^(-λt) describes the amount of a radioactive substance N at time t, where N₀ is the initial amount, λ is the decay constant, and e is the base of the natural logarithm. Determining half-lives or decay rates requires solving exponential equations.

  4. Physics: Many physical phenomena, such as the discharge of a capacitor in an RC circuit, are modeled by exponential equations. These equations are used to describe how quantities change over time, and solving them is essential for understanding the behavior of these systems.

  5. Biology: Exponential functions are used to model biological processes, such as the growth of bacterial cultures or the spread of infectious diseases. These models help scientists make predictions and understand the dynamics of these processes.

In conclusion, solving for x in the equation 3^x = 12 is a quintessential example of how logarithms are used to tackle exponential equations. By understanding the properties of logarithms and applying them correctly, we can transform complex exponential problems into straightforward algebraic solutions. We have demonstrated two methods, using both common and natural logarithms, to arrive at the solution x ≈ 2.262. Verifying the solution and being aware of common mistakes are crucial steps in the problem-solving process.

Furthermore, we’ve highlighted the broad applicability of exponential equations in various real-world scenarios, emphasizing the importance of mastering these mathematical concepts. Whether it’s calculating compound interest in finance, modeling population growth, or understanding radioactive decay, exponential equations play a vital role in many scientific and practical applications. By mastering the techniques to solve these equations, you are equipping yourself with a valuable tool for analyzing and understanding the world around you. Keep practicing, stay curious, and you'll continue to excel in mathematics! Understanding logarithms and exponential functions is key to unlocking many mathematical challenges.