Solving Logarithmic Equations: Graphing The Solution

Hey guys! Today, we're diving into the world of logarithmic equations and how to visualize their solutions on a graph. Specifically, we're going to tackle the equation $\log _2(3 x-1)=2$ and break down each step to understand not only how to solve it, but also why the solution makes sense.

Understanding the Logarithmic Equation

Before we jump into solving and graphing, let's make sure we're all on the same page about what a logarithmic equation actually means. The equation $\log _2(3 x-1)=2$ is essentially asking: "To what power must we raise 2 to get (3x-1)?" In this case, the answer is 2, because the equation states that the logarithm base 2 of (3x-1) equals 2. Logarithms are just the inverse of exponentiation, which means they "undo" each other. This is super useful when we want to solve for a variable that's stuck inside an exponent or, in this case, inside a logarithm.

Now, you might be wondering, "Why are logarithms even important?" Well, they show up all over the place in math and science! They're used to model things like the growth of populations, the decay of radioactive materials, the intensity of earthquakes (that's the Richter scale!), and even the loudness of sound (decibels). Understanding logarithms opens up a whole new toolbox for analyzing and understanding the world around us. Plus, being able to solve logarithmic equations is a fundamental skill in algebra and calculus, so it's definitely worth mastering.

Think of it like this: exponentiation takes a base and an exponent and gives you a result. Logarithms take the base and the result and give you the exponent. They're two sides of the same coin! So, when you see a logarithmic equation, try to think about the underlying exponential relationship. It can make things a lot clearer. Remember, the goal is to isolate the variable, and logarithms provide a way to "unwrap" it from the exponent. By understanding this relationship, you can confidently approach and solve logarithmic equations.

Solving the Equation $\log _2(3 x-1)=2$

Okay, let's get our hands dirty and actually solve the equation. Our mission is to isolate 'x'. To do that, we need to "undo" the logarithm. The key here is to remember the relationship between logarithms and exponents that we discussed earlier. Since the logarithm is base 2, we can rewrite the equation in exponential form using the following logic:

If $\log _b(a) = c$, then $b^c = a$.

Applying this to our equation, $\log _2(3 x-1)=2$, we get:

$2^2 = 3x - 1$

See what we did there? We took the base of the logarithm (2), raised it to the power of the result (2), and set it equal to the argument of the logarithm (3x - 1). Now we have a much simpler equation to solve! Next, let's simplify the left side of the equation:

$4 = 3x - 1$

Now, we need to isolate 'x'. First, let's add 1 to both sides of the equation to get rid of the -1 on the right side:

$4 + 1 = 3x - 1 + 1$

$5 = 3x$

Finally, to get 'x' by itself, we'll divide both sides of the equation by 3:

$\frac{5}{3} = \frac{3x}{3}$

$x = \frac{5}{3}$

So, there you have it! The solution to the equation $\log _2(3 x-1)=2$ is $x = \frac{5}{3}$. But wait, we're not done yet! We need to make sure this solution is valid.

Verifying the Solution

It's super important to check our solution, especially when dealing with logarithmic equations. Why? Because logarithms are only defined for positive arguments. In other words, you can only take the logarithm of a positive number. If we plug our solution back into the original equation and the argument of the logarithm becomes negative or zero, then our solution is not valid.

Let's plug $x = \frac{5}{3}$ back into the original equation, $\log _2(3 x-1)=2$:

$\log _2(3(\frac{5}{3})-1)=2$

$\log _2(5-1)=2$

$\log _2(4)=2$

Is this true? Yes! Because $2^2 = 4$. So, our solution $x = \frac{5}{3}$ is indeed valid.

Graphing the Solution

Now comes the fun part: visualizing the solution on a graph! When we talk about graphing the solution to an equation like this, we're essentially looking for the point where the graph of the function $y = \log _2(3x - 1)$ intersects the horizontal line $y = 2$. In other words, we're looking for the x-value where the y-value of the logarithmic function equals 2.

Unfortunately, I can't actually draw a graph for you here. But I can describe what it would look like and how to interpret it. The graph of $y = \log _2(3x - 1)$ is a logarithmic curve that starts very close to the vertical line $x = \frac{1}{3}$ (this is a vertical asymptote, because the argument of the logarithm, 3x - 1, must be greater than 0) and increases slowly as x increases. The horizontal line $y = 2$ is simply a straight line that runs horizontally at a height of 2 on the y-axis.

The point where these two graphs intersect is the solution to our equation. If you were to plot these two graphs, you would see that they intersect at the point $(\frac{5}{3}, 2)$. This visually confirms that $x = \frac{5}{3}$ is the solution to the equation $\log _2(3 x-1)=2$.

To find the correct graph, look for a logarithmic function that has a vertical asymptote at $x = \frac{1}{3}$ and intersects the line $y = 2$ at $x = \frac{5}{3}$. The graph should show the curve approaching the vertical asymptote but never touching it, and then gradually increasing and crossing the horizontal line at the point we calculated.

Remember: graphing is a powerful tool for visualizing solutions and understanding the behavior of functions. In this case, it helps us see how the logarithmic function and the constant value of 2 relate to each other, and it provides a visual confirmation of our algebraic solution. Keep an eye out for the key features of the graph, like the vertical asymptote and the intersection point, to correctly identify the solution.

Conclusion

So, to wrap it all up, we successfully solved the equation $\log _2(3 x-1)=2$ and found the solution to be $x = \frac{5}{3}$. We verified the solution to make sure it was valid, and we discussed how to visualize the solution on a graph by finding the intersection point of the logarithmic function and the horizontal line $y = 2$. Remember the key steps: rewrite the logarithmic equation in exponential form, solve for the variable, and verify your solution. And don't forget the power of graphing to visualize and confirm your results!

Keep practicing, and you'll become a logarithm master in no time! You got this!