Solve `3 Log (3-x) = 1-x` With A Graphing Calculator

Hey there, math explorers! Ever run into an equation that just makes your brain hurt trying to solve it by hand? You know, those tricky beasts that mix different types of functions, like logarithms and plain old linear terms? Well, today, we're diving deep into one such puzzle: 3 log (3-x) = 1-x. Trying to solve this specific equation algebraically is like trying to untie a knot with your teeth – it's possible, but incredibly difficult and usually not worth the effort. That's where your trusty graphing calculator becomes your absolute superpower! This amazing tool isn't just for plotting pretty curves; it's designed to help us visualize mathematical relationships and, crucially, find solutions to equations that would otherwise be beyond our reach without advanced calculus or numerical methods. We're talking about equations where a log term (which behaves in a specific, non-linear way) is jammed up against a simple x term. Our mission today is to show you, step-by-step, how to leverage the power of your graphing calculator, whether it's a physical TI-84 or a fantastic online tool like Desmos, to conquer this problem. We'll walk through everything from setting up the functions correctly, understanding the domain restrictions that come with logarithms, to skillfully finding those elusive intersection points and, finally, accurately rounding your answers to the nearest hundredth. So, grab your calculator (or open a new browser tab for Desmos!), because we're about to make solving complex equations not just easy, but actually fun.

Understanding the Equation: 3 log (3-x) = 1-x

Alright, let's break down the main event, our equation: 3 log (3-x) = 1-x. Before we even touch a graphing calculator, it's super important to understand what we're dealing with. On the left side, we have 3 log (3-x). This involves a logarithm, specifically the common logarithm (base 10, unless otherwise specified, though some calculators default to natural log, so always double-check!). The crucial thing about logarithms is their domain restriction. You can never take the logarithm of zero or a negative number. This means that whatever is inside the parentheses, (3-x), must be greater than zero. So, 3-x > 0, which simplifies to x < 3. This little detail is a huge deal, guys, because it tells us that any solution we find must be less than 3. If your calculator spits out an x value of, say, 5, you'll know instantly that something is off, or it's an extraneous solution that doesn't fit the log's domain. On the right side, we have 1-x, which is a straightforward linear function. It's a simple straight line that goes down as x increases. The challenge arises because we have a logarithmic curve trying to intersect with a straight line. There's no neat algebraic trick to isolate x when it's stuck inside a logarithm on one side and free as a bird on the other. This is precisely why we turn to graphical methods. The core idea for using a graphing calculator is to transform our single equation into a system of two separate functions: y1 = 3 log (3-x) and y2 = 1-x. We're essentially asking: At what x value (or values) do these two functions produce the same y value? Those points of equality are exactly where the graphs of y1 and y2 intersect. By visualizing these two functions, we can visually pinpoint their meeting places, and our calculator will do the heavy lifting of finding the precise coordinates for us. Keeping that x < 3 domain in mind will be super helpful when setting up our viewing window and interpreting our results, making sure we're looking for valid solutions within the bounds of the logarithmic function.

Setting Up Your Graphing Calculator: A Step-by-Step Guide

Now for the fun part: getting these equations onto your graphing calculator! Whether you're rocking an old-school TI-84 or cruising with a sleek online tool like Desmos, the principle is the same: enter two functions and watch 'em graph. Let's start with the classic hardware.

Getting Started with a TI-84 (or similar)

For those of you with a physical graphing calculator like the TI-83 or TI-84 Plus, here’s your roadmap to success. First things first, turn that baby on! Once it's fired up, you'll want to access the Y= editor. You usually find this button on the top left of your keypad. Press it, and you'll see a list: Y1=, Y2=, Y3=, and so on. This is where we'll input our two functions. For Y1, we'll enter the left side of our equation, and for Y2, the right side.

Step 1: Input Y1 = 3 log (3-x)

• Go to Y1=.
• Type 3.
• Find the LOG button. It's usually near the 7 key. Press it.
• Open a parenthesis: (.
• Type 3 - X. The X button is typically X,T,theta,n.
• Close the parenthesis: ).
• So, Y1 should look like: 3log(3-X). Pro-tip: Make sure you're using the correct log button. Most common calculators default to base 10 for log. If your equation specified natural log (ln), you'd use the LN button instead. But for this problem, log base 10 is standard.

Step 2: Input Y2 = 1-x

• Go down to Y2=.
• Type 1 - X.
• So, Y2 should look like: 1-X.

Step 3: Setting the Window – This is CRUCIAL!

This is where many people get stuck. If your window isn't set correctly, you might not see the intersection points, or even the graphs themselves! Remember our domain restriction? x < 3. This is a huge hint. Our Xmax shouldn't be much larger than 3. Let's try some intelligent guesses:

• Press the WINDOW button.
• Xmin: Let's go a bit lower than where we expect intersections might be, maybe -5.
• Xmax: Since x must be less than 3, let's set Xmax = 5 (giving us a bit of breathing room to see the asymptote, though Xmax=3 would be fine too, but 5 lets us see the cutoff).
• Xscl: This is how often tick marks appear on the x-axis. 1 is usually fine.
• Ymin: The linear function 1-x can go pretty low as x gets positive. The log function also tends to grow slowly. Let's try Ymin = -5.
• Ymax: For 1-x when x is small (e.g., x=-5), y would be 1 - (-5) = 6. For the log function, 3 log(3-(-5)) = 3 log(8) approx 3 * 0.9 = 2.7. So, Ymax = 10 should give us enough vertical range.
• Yscl: 1 is fine.

Self-correction tip: If you graph and don't see anything, or only one curve, adjust your window! A common mistake is not considering the domain of the log function or simply having a window that's too zoomed in or out. The ZOOM menu also has options like ZoomFit (option 0) which can sometimes help, but manual adjustment based on function understanding is often best for these specific types of problems. For this particular equation, x values could be between -10 and 3, and y values could range from -5 to 5 or 10. A good starting point often is Xmin = -10, Xmax = 3.5, Ymin = -5, Ymax = 5 for a tight view near the critical region of the log function.

Step 4: Graph it!

• Now, press the GRAPH button. You should see both your logarithmic curve and your straight line appear. Hopefully, you'll see them cross each other! Don't fret if you don't immediately; just go back to WINDOW and tweak your Xmin, Xmax, Ymin, and Ymax settings until both graphs are clearly visible and any potential intersection points are within view. Often, a good strategy is to use the ZOOM Standard (option 6 in ZOOM) first to get a general idea, and then ZOOM Out (option 3) or manually adjust the window to ensure you capture everything.

Using Online Tools Like Desmos for Super Easy Graphing

If you prefer a more modern, visually intuitive approach, or if you don't have a physical calculator handy, Desmos is your absolute best friend. It’s a free online graphing calculator that works wonders on any device with a web browser. Seriously, guys, if you haven't tried Desmos, you're missing out! Its user interface is incredibly straightforward, and it handles most of the tedious window adjustments automatically.

Step 1: Head to Desmos.com/calculator

• Open your web browser and navigate to www.desmos.com/calculator.

Step 2: Input Your Functions

• You'll see a blank expression box on the left. Simply type your first function: y = 3 log (3-x). Desmos is smart enough to recognize log as base 10 by default. As you type, the graph will appear instantly on the right! It’s magical. Desmos even subtly shades the region where the logarithm is undefined (i.e., x >= 3), making the domain restriction visually clear without you having to think about it too hard. This feature is incredibly helpful for understanding the behavior of functions. You can use the standard keyboard for numbers and x, and the log function is easily accessible by typing log or by using the functions keyboard that pops up.
• Click on the + sign or hit enter to add another expression line.
• Type your second function: y = 1-x. Again, the line will appear immediately.

That's it for setup on Desmos! No need to fuss with window settings initially, as Desmos automatically scales and zooms to show relevant parts of your graphs. You can easily pinch-to-zoom or drag the graph around with your mouse or trackpad if you want to explore different regions. The beauty of Desmos is its instant feedback; you literally see the graph forming as you type each character. This makes experimenting with different functions and understanding their shapes incredibly easy and intuitive. It also visually represents the domain x < 3 by simply not drawing the log function for x values of 3 or greater, providing an immediate visual confirmation of our earlier domain analysis. This ease of use makes Desmos an excellent tool for quick checks and for those who are just starting out with graphical solutions, reducing the initial learning curve associated with traditional graphing calculators.

Finding the Solutions: Intersection Points are Key!

Once you have your graphs displayed, whether on a TI-84 or Desmos, the next critical step is to find where they cross. These intersection points are the solutions to our equation!

On the TI-84: The "CALC" Menu and "Intersect" Feature

Your TI-84 has a dedicated feature for finding these points of intersection. It's usually found under the CALC menu, which is accessed by pressing 2nd then TRACE.

Step 1: Access the CALC menu

• Press 2nd (the blue or yellow button) then TRACE (which is typically labeled CALC above it).
• A menu will pop up. You'll want to select option 5: intersect. Use the arrow keys to navigate down to 5 and press ENTER.

Step 2: Identify the First Curve

• Your calculator will now show the graph and prompt you: "First curve?". You'll see the cursor blinking on one of your functions. If it's on Y1 = 3 log (3-x), press ENTER. If it's on the wrong curve, use the UP or DOWN arrow keys to move the cursor to Y1 and then press ENTER.

Step 3: Identify the Second Curve

• Next, it will prompt: "Second curve?". The cursor will jump to the other function, Y2 = 1-x. Again, if it's correct, press ENTER. If not, use the UP or DOWN arrows to select Y2 and press ENTER.

Step 4: Make a Guess

• Finally, it will ask: "Guess?". This step is important, especially if your graphs intersect at more than one point. Use the LEFT or RIGHT arrow keys to move the cursor close to the intersection point you want to find. Once the cursor is near the intersection, press ENTER. The calculator will then display the coordinates (X= and Y=) of that intersection point.

Let's apply this to our equation 3 log (3-x) = 1-x. After graphing, you should observe two intersection points. You'll need to repeat the