Math Problems: Logarithms & Quadratic Functions Explained

Hey guys! Let's dive into some math problems today. We'll be tackling a logarithm problem and then moving on to find the coordinates where a quadratic function intersects the X-axis. Don't worry, it's not as scary as it sounds. We'll break it down step by step. Get ready to flex those math muscles! We are going to use our skills and knowledge to solve the problems.

Logarithm Problem: ${}^2\log 81 - {}^2\log 27 + {}^2\log 9 = \dots$

Alright, let's get started with the logarithm problem. The question is: ${}^2\log 81 - {}^2\log 27 + {}^2\log 9 = \dots$. This might look intimidating at first, but trust me, it's manageable. We'll use the properties of logarithms to simplify this expression and find the solution. Remember, the key to solving these types of problems is to understand the properties of logarithms and how to apply them correctly. We will dissect the question.

First, let's refresh our memory on the basic properties of logarithms. There are a few key rules that will be super helpful here:

• Product Rule: ${}^a\log(b * c) = {}^a\log b + {}^a\log c$
• Quotient Rule: ${}^a\log(b / c) = {}^a\log b - {}^a\log c$
• Power Rule: ${}^a\log(b^c) = c * {}^a\log b$

Now, let's get back to our problem: ${}^2\log 81 - {}^2\log 27 + {}^2\log 9$. Notice that all the terms have the same base, which is 2. This makes our lives a bit easier. Our main goal is to simplify the expression using the logarithm properties. We will work with the number to determine what steps to take. We can start by simplifying each term individually. The first term is ${}^2\log 81$. We know that $81 = 3^4$, so we can rewrite this term using the power rule: ${}^2\log 81 = {}^2\log(3^4) = 4 * {}^2\log 3$. Next, we have ${}^2\log 27$. Since $27 = 3^3$, we can rewrite this as: ${}^2\log 27 = {}^2\log(3^3) = 3 * {}^2\log 3$. Finally, we have ${}^2\log 9$. Because $9 = 3^2$, this becomes: ${}^2\log 9 = {}^2\log(3^2) = 2 * {}^2\log 3$. Now, substitute these simplified terms back into our original equation: $4 * {}^2\log 3 - 3 * {}^2\log 3 + 2 * {}^2\log 3$. Combine the terms, since they all have the same logarithmic component: $(4 - 3 + 2) * {}^2\log 3 = 3 * {}^2\log 3$. So, the answer is $3 * {}^2\log 3$. Therefore, the correct answer is E. $3{}^2 \log 3$. That wasn't too bad, right? Keep practicing, and you'll become a logarithm master in no time! We have successfully solved the first problem.

Finding the Intersection of a Quadratic Function with the X-axis

Now, let's move on to the second part of our problem, which deals with quadratic functions. The question asks us to find the coordinates where the graph of the quadratic function $f(x) = 3x^2 + 9x + 6$ intersects the X-axis. This is a classic problem. Let's break down the method to find this intersection point. Do not worry, it is going to be simpler than you think.

To find where a function intersects the X-axis, we need to find the values of x where f(x) = 0. Basically, we're looking for the x-intercepts of the graph. So, we need to solve the equation $3x^2 + 9x + 6 = 0$. This is a quadratic equation, and there are several ways to solve it: factoring, completing the square, or using the quadratic formula. Let's go with factoring because it is often the easiest method if the equation can be factored. Also, the quadratic equation is the most used equation in our daily life.

First, let's see if we can simplify the equation by dividing all the terms by a common factor. In this case, all the coefficients (3, 9, and 6) are divisible by 3. Divide the entire equation by 3: $(3x^2 + 9x + 6) / 3 = 0 / 3$, which simplifies to $x^2 + 3x + 2 = 0$. Now, let's try to factor this quadratic equation. We're looking for two numbers that multiply to give us 2 (the constant term) and add up to 3 (the coefficient of the x term). Those two numbers are 1 and 2. So, we can factor the equation as $(x + 1)(x + 2) = 0$. Setting each factor equal to zero, we get two possible solutions for x: $x + 1 = 0$ or $x + 2 = 0$. Solving for x in each case gives us $x = -1$ and $x = -2$. These are the x-coordinates of the points where the graph intersects the X-axis. To find the coordinates, we need to remember that the Y-coordinate at the X-axis is always 0. Therefore, the intersection points are (-1, 0) and (-2, 0). So, the graph of the function $f(x) = 3x^2 + 9x + 6$ intersects the X-axis at the points (-1, 0) and (-2, 0). That's it! We have successfully solved the second problem.

Summary and Tips

Alright, guys, we've successfully tackled both problems. Let's recap what we've learned:

• Logarithms: We used the properties of logarithms (product, quotient, and power rules) to simplify a logarithmic expression. Remember to always look for opportunities to rewrite terms using these properties.
• Quadratic Functions: We found the X-intercepts of a quadratic function by setting f(x) = 0 and solving the resulting quadratic equation. We used factoring in this case, but remember that completing the square or the quadratic formula are also viable methods.

Here are some extra tips to help you with similar problems:

• Practice, Practice, Practice: The more you practice, the more familiar you'll become with the different types of problems and the techniques to solve them.
• Know Your Formulas: Make sure you know the basic formulas and properties for logarithms and quadratic equations. You will use this in all aspects of your life.
• Break it Down: When faced with a complex problem, break it down into smaller, more manageable steps. This will make the problem less intimidating and easier to solve.
• Check Your Work: Always double-check your answers to make sure they make sense and that you haven't made any calculation errors.

Keep up the great work, and don't be afraid to ask for help if you get stuck. Math can be fun, and with practice, you can master these concepts. Keep up the great work and never stop learning.