Solving The Logarithmic Equation Ln(3) - Ln(3-9x) = Ln(5)
Hey everyone! Today, we're diving into a fascinating mathematical problem: solving a logarithmic equation. Logarithmic equations might seem a bit intimidating at first, but don't worry, we'll break it down step by step. Our mission is to find all the solutions to the equation: ln(3) - ln(3 - 9x) = ln(5). If there's more than one solution, we'll present our answers as a comma-separated list. So, grab your thinking caps, and let's get started!
Understanding Logarithms
Before we jump into solving the equation, let's have a quick refresher on logarithms. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, if we have an equation like b^y = x, the logarithm (base b) of x is y, written as log_b(x) = y. The natural logarithm, denoted as "ln," is a logarithm with the base e (Euler's number, approximately 2.71828). So, ln(x) is the same as log_e(x). Understanding this fundamental concept is crucial for tackling logarithmic equations effectively. Logarithms help us solve equations where the unknown variable is in the exponent, and they are widely used in various fields like physics, engineering, and computer science. In our case, we're dealing with natural logarithms, which have properties that make them particularly useful for solving equations involving exponential decay or growth. Remember, the logarithm of a number represents the power to which the base must be raised to equal that number. This relationship is the key to manipulating and simplifying logarithmic expressions. Let's keep this in mind as we proceed to solve our equation!
Properties of Logarithms
To effectively solve our equation, we need to utilize some key properties of logarithms. These properties allow us to manipulate logarithmic expressions and simplify them into a form that's easier to work with. The main property we'll be using here is the quotient rule of logarithms, which states that the logarithm of a quotient is equal to the difference of the logarithms. Mathematically, this is expressed as: ln(a) - ln(b) = ln(a/b). This property is extremely useful for combining logarithmic terms and simplifying equations. Another important property is that if ln(a) = ln(b), then a = b. This allows us to eliminate the logarithms once we've simplified the equation to a point where we have a single logarithm on each side. In addition to the quotient rule, there's also the product rule, which states that ln(a) + ln(b) = ln(ab), and the power rule, which states that ln(a^k) = k * ln(a). While we won't be directly using the product and power rules in this particular problem, they are essential tools in your logarithmic equation-solving arsenal. Mastering these properties will empower you to tackle a wide range of logarithmic problems with confidence. Now, let's apply these concepts to our specific equation and see how they help us find the solution!
Solving the Equation Step-by-Step
Alright, guys, let's dive into solving the equation ln(3) - ln(3 - 9x) = ln(5). The first thing we're going to do is use the quotient rule of logarithms to combine the left side of the equation. Remember, the quotient rule states that ln(a) - ln(b) = ln(a/b). Applying this to our equation, we get:
ln(3 / (3 - 9x)) = ln(5)
Now that we have a single logarithm on each side of the equation, we can use the property that if ln(a) = ln(b), then a = b. This allows us to eliminate the logarithms and simplify the equation further:
3 / (3 - 9x) = 5
Next, we need to get rid of the fraction. We can do this by multiplying both sides of the equation by (3 - 9x):
3 = 5(3 - 9x)
Now, let's distribute the 5 on the right side:
3 = 15 - 45x
We want to isolate the term with x, so let's subtract 15 from both sides:
3 - 15 = -45x
-12 = -45x
Finally, to solve for x, we'll divide both sides by -45:
x = -12 / -45
x = 4 / 15
So, we've found a potential solution: x = 4/15. But hold on, we're not done yet! We need to check if this solution is valid.
Checking for Extraneous Solutions
Okay, we've arrived at a potential solution, x = 4/15. But here's a crucial step we can't skip: checking for extraneous solutions. Extraneous solutions are values that we obtain during the solving process that don't actually satisfy the original equation. This often happens with logarithmic equations because the logarithm function is only defined for positive arguments. So, we need to make sure that plugging x = 4/15 back into the original equation doesn't result in taking the logarithm of a negative number or zero.
Our original equation was: ln(3) - ln(3 - 9x) = ln(5)
The only term we need to worry about is ln(3 - 9x). We need to ensure that (3 - 9x) is greater than zero. Let's plug in our solution, x = 4/15:
3 - 9(4/15) = 3 - (36/15) = 3 - (12/5) = (15/5) - (12/5) = 3/5
Since 3/5 is a positive number, our solution x = 4/15 is valid! If we had obtained a negative number or zero, we would have had to discard that solution. Checking for extraneous solutions is a vital part of solving logarithmic equations, ensuring that our answer is not only mathematically correct but also makes sense within the context of the original problem. Now that we've confirmed our solution, we can confidently state the final answer.
Final Answer
Phew! We've gone through the entire process, from understanding logarithms and their properties to solving the equation and checking for extraneous solutions. We started with ln(3) - ln(3 - 9x) = ln(5), and after applying the quotient rule, simplifying, and solving for x, we arrived at the solution x = 4/15. We then meticulously checked this solution to ensure it didn't lead to taking the logarithm of a non-positive number, confirming its validity. So, the final answer to our equation is:
x = 4/15
Great job, guys! We successfully navigated this logarithmic equation. Remember, the key to solving these types of problems is understanding the properties of logarithms, carefully applying them, and always, always checking for extraneous solutions. Keep practicing, and you'll become a logarithm-solving pro in no time! If you have any more questions or want to explore other mathematical challenges, feel free to ask. Let's keep the learning going! This was a fun problem to solve, and I hope you found the step-by-step explanation helpful. Happy math-ing!
