Solving 4^(x-3) = 18 A Step-by-Step Guide
Introduction
In this comprehensive guide, we will tackle the exponential equation 4^(x-3) = 18 and find the solution for x, rounded to the nearest thousandth. Exponential equations are a fundamental topic in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and finance. We will walk through the process step by step, utilizing logarithmic properties to isolate x and arrive at the final answer. Whether you're a student learning algebra or someone looking to brush up on your math skills, this article will provide a clear and detailed explanation of how to solve this type of equation.
Understanding Exponential Equations
Before diving into the solution, let's briefly discuss what exponential equations are. An exponential equation is an equation in which the variable appears in the exponent. The general form of an exponential equation is a^x = b, where a is the base, x is the exponent, and b is the result. Solving exponential equations often involves using logarithms, which are the inverse operation of exponentiation. Logarithms allow us to bring the exponent down and solve for the variable. Understanding the relationship between exponential functions and logarithmic functions is key to mastering these types of problems. This article aims to make this understanding clearer by applying it to the specific equation at hand.
The Importance of Logarithms
Logarithms are essential tools for solving exponential equations because they allow us to isolate the variable in the exponent. The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. For instance, the logarithm of 100 to the base 10 is 2 because 10^2 = 100. In mathematical notation, this is written as log10(100) = 2. Different bases can be used for logarithms, with the most common being base 10 (common logarithm) and base e (natural logarithm), where e is approximately 2.71828. The natural logarithm is denoted as ln(x). Understanding and applying the properties of logarithms is crucial for solving exponential equations accurately and efficiently. This guide will show you how to apply these properties to solve the given equation.
Step-by-Step Solution
1. Take the Logarithm of Both Sides
The first step in solving the equation 4^(x-3) = 18 is to take the logarithm of both sides. We can use either the common logarithm (base 10) or the natural logarithm (base e). For this example, we will use the natural logarithm, as it often simplifies calculations in calculus and other advanced mathematical contexts. Taking the natural logarithm of both sides, we get:
ln(4^(x-3)) = ln(18)
This step is crucial because it allows us to use the power rule of logarithms, which will help us bring down the exponent.
2. Apply the Power Rule of Logarithms
The power rule of logarithms states that ln(a^b) = b * ln(a). Applying this rule to the left side of our equation, we get:
(x - 3) * ln(4) = ln(18)
This step is a key transformation because it moves the variable x from the exponent to a linear term, making it easier to isolate. By applying the power rule, we've converted an exponential equation into a linear equation involving logarithms. This is a common strategy in solving exponential equations, and mastering this step is essential for tackling more complex problems.
3. Isolate (x - 3)
Now, we need to isolate the term (x - 3). To do this, we divide both sides of the equation by ln(4):
x - 3 = ln(18) / ln(4)
This step involves a simple algebraic manipulation, but it's a crucial one for solving for x. By dividing both sides by ln(4), we have successfully isolated the term containing x. This brings us closer to our goal of finding the value of x. The next step will involve adding 3 to both sides to completely isolate x.
4. Solve for x
To solve for x, we add 3 to both sides of the equation:
x = (ln(18) / ln(4)) + 3
This step completes the algebraic isolation of x. Now we have an expression for x in terms of natural logarithms. To find the numerical value of x, we will need to evaluate the logarithms and perform the arithmetic operations. This is where a calculator comes in handy, as it can provide accurate values for logarithms.
5. Calculate the Value of x
Using a calculator, we find the approximate values of ln(18) and ln(4):
ln(18) ≈ 2.89037
ln(4) ≈ 1.38629
Now, we substitute these values into our equation:
x ≈ (2.89037 / 1.38629) + 3
x ≈ 2.08516 + 3
x ≈ 5.08516
This calculation gives us an approximate value for x. However, we still need to round it to the nearest thousandth as requested in the problem statement.
6. Round to the Nearest Thousandth
Rounding 5.08516 to the nearest thousandth, we look at the digit in the ten-thousandths place, which is 1. Since 1 is less than 5, we round down:
x ≈ 5.085
Therefore, the solution to the equation 4^(x-3) = 18, rounded to the nearest thousandth, is approximately 5.085. This final step ensures that our answer meets the specific requirement of the problem, which is to provide the solution rounded to a certain decimal place.
Alternative Method: Using Base 10 Logarithms
While we used natural logarithms in our solution, we could have also used common logarithms (base 10). The process is essentially the same, but it's helpful to see it done both ways to reinforce the concept. Let's revisit the equation 4^(x-3) = 18 and solve it using base 10 logarithms.
1. Take the Base 10 Logarithm of Both Sides
Taking the base 10 logarithm of both sides, we get:
log10(4^(x-3)) = log10(18)
2. Apply the Power Rule of Logarithms
Using the power rule, we have:
(x - 3) * log10(4) = log10(18)
3. Isolate (x - 3)
Divide both sides by log10(4):
x - 3 = log10(18) / log10(4)
4. Solve for x
Add 3 to both sides:
x = (log10(18) / log10(4)) + 3
5. Calculate the Value of x
Using a calculator, we find the approximate values of log10(18) and log10(4):
log10(18) ≈ 1.25527
log10(4) ≈ 0.60206
Substitute these values into our equation:
x ≈ (1.25527 / 0.60206) + 3
x ≈ 2.08516 + 3
x ≈ 5.08516
6. Round to the Nearest Thousandth
Rounding 5.08516 to the nearest thousandth, we get:
x ≈ 5.085
As you can see, we arrive at the same answer using base 10 logarithms as we did with natural logarithms. This demonstrates that the choice of logarithm base does not affect the final solution, as long as the logarithm properties are applied correctly.
Conclusion
In this guide, we successfully solved the exponential equation 4^(x-3) = 18 and found that x is approximately 5.085 when rounded to the nearest thousandth. We walked through each step of the process, explaining the reasoning behind each operation and highlighting the importance of logarithmic properties. By taking the logarithm of both sides, applying the power rule, and isolating the variable, we were able to transform the exponential equation into a linear equation that we could easily solve. We also demonstrated that both natural logarithms and common logarithms can be used to solve these types of equations, providing flexibility in your approach. Understanding these techniques is essential for solving exponential equations and mastering algebra. Practice is key to solidifying these skills, so be sure to work through additional examples to build your confidence and proficiency. Remember, the key to solving exponential equations lies in understanding logarithms and their properties, and with practice, you can become adept at tackling these problems with ease.
