Solving Logarithmic Equations: A Step-by-Step Guide

Nov 6, 2025 by Admin 52 views

Hey guys! Let's dive into solving logarithmic equations. Logarithmic equations might seem intimidating at first, but trust me, they're not as scary as they look. We'll break down the equation $\log _2(9 x)-\log _2 3=3$ step-by-step so you can understand exactly how to tackle these problems. So, grab your pencils, and let's get started!

Understanding Logarithms

Before we jump into the solution, let's quickly recap what logarithms are. Logarithms are essentially the inverse operation of exponentiation. Think of it this way: if $2^3 = 8$ , then $\log_2 8 = 3$ . The logarithm tells you what exponent you need to raise the base (in this case, 2) to get a certain number (in this case, 8). The key here is understanding the relationship between exponents and logarithms. This is crucial for manipulating and solving logarithmic equations effectively. When dealing with logarithms, there are a few key properties you need to know. For example, the product rule states that the logarithm of a product is equal to the sum of the logarithms: $\log_b(mn) = \log_b m + \log_b n$ . Conversely, the quotient rule (which we'll use in our problem) says that the logarithm of a quotient is equal to the difference of the logarithms: $\log_b(\frac{m}{n}) = \log_b m - \log_b n$ . Finally, the power rule states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number: $\log_b(m^p) = p \log_b m$ . Mastering these properties is essential for simplifying logarithmic expressions and solving equations. These rules allow us to combine, separate, and manipulate logarithmic terms, making the equation easier to solve. Understanding these properties is like having the right tools in your toolbox – they allow you to approach various types of logarithmic problems with confidence. So, before moving forward, make sure you're comfortable with these rules. If needed, take a moment to review them. Trust me, it will make the rest of the process much smoother. Remember, practice makes perfect, so try applying these properties to different logarithmic expressions to solidify your understanding.

Step-by-Step Solution

Okay, let's get to the fun part – solving the equation $\log _2(9 x)-\log _2 3=3$ . Here's how we can break it down:

1. Apply the Quotient Rule

Remember the quotient rule? It states that $\log_b(\frac{m}{n}) = \log_b m - \log_b n$ . We can use this in reverse to combine the two logarithmic terms on the left side of our equation.
Applying the quotient rule, we get: $\log _2(\frac{9x}{3}) = 3$ .

2. Simplify the Fraction

Inside the logarithm, we have the fraction $\frac{9x}{3}$ . This can be simplified quite easily.
Dividing 9x by 3, we get: $\log _2(3x) = 3$ .

3. Convert to Exponential Form

Now, let's get rid of the logarithm! Remember the relationship between logarithms and exponents? If $\log_b a = c$ , then $b^c = a$ .
Applying this to our equation, we get: $2^3 = 3x$ .

4. Evaluate the Exponential Term

We have $2^3$ on the left side. Let's calculate that.
$2^3$ is simply 2 multiplied by itself three times: $2 * 2 * 2 = 8$ . So, our equation becomes: $8 = 3x$ .

5. Isolate x

We're almost there! To solve for x, we need to get it by itself.
Divide both sides of the equation by 3: $\frac{8}{3} = x$ .

6. The Solution

We've found our solution! x = 8/3.
So, none of the options you provided (A. $x=\frac{3}{8}$ , B. $x=\frac{0}{3}$ , C. $x=3$ ) are correct. The correct answer is x = 8/3.

Common Mistakes to Avoid

When solving logarithmic equations, there are a few common pitfalls to watch out for. One mistake is misapplying the logarithmic properties, such as the product, quotient, and power rules. It's crucial to remember these rules correctly and apply them in the appropriate situations. For instance, confusing the quotient rule with the product rule can lead to incorrect simplifications and ultimately, the wrong answer. Another frequent error is forgetting to check for extraneous solutions. Since logarithms are only defined for positive arguments, it's essential to ensure that the solutions you obtain don't result in taking the logarithm of a negative number or zero. This often involves plugging the solutions back into the original equation to verify their validity. Ignoring this step can lead to including solutions that are not actually valid. Additionally, some students struggle with converting between logarithmic and exponential forms. This conversion is a fundamental step in solving many logarithmic equations, so it's important to practice and become comfortable with it. A clear understanding of the relationship between logarithms and exponents is key to making this conversion accurately. Finally, another common mistake is making algebraic errors during the simplification process. This can include errors in arithmetic, such as incorrect multiplication or division, as well as errors in algebraic manipulation, such as incorrectly combining like terms. To avoid these mistakes, it's helpful to double-check your work and pay close attention to each step in the process. Practice and attention to detail are your best tools for avoiding these common errors.

Practice Problems

To really solidify your understanding, let's try a couple of practice problems. Working through these examples will help you become more comfortable with the steps involved in solving logarithmic equations. The more you practice, the more natural these techniques will become. Feel free to grab a pen and paper and work through these problems on your own. Then, compare your solutions to the ones provided below. Remember, the key is to apply the logarithmic properties and algebraic manipulations we've discussed. If you get stuck, don't worry! Just review the steps and try again. Practice makes perfect, and each problem you solve will build your confidence and skills. So, let's dive in and put your knowledge to the test!

Problem 1

Solve for x: $\log_3(2x + 1) = 2$

Problem 2

Solve for x: $\log_4(x) + \log_4(x - 6) = 2$

Solutions to Practice Problems

Problem 1 Solution

Convert to exponential form: $3^2 = 2x + 1$
Simplify: $9 = 2x + 1$
Subtract 1 from both sides: $8 = 2x$
Divide by 2: $x = 4$

Problem 2 Solution

Use the product rule to combine the logarithms: $\log_4(x(x - 6)) = 2$
Convert to exponential form: $4^2 = x(x - 6)$
Simplify: $16 = x^2 - 6x$
Rearrange into a quadratic equation: $x^2 - 6x - 16 = 0$
Factor the quadratic: $(x - 8)(x + 2) = 0$
Solve for x: $x = 8$ or $x = -2$
Check for extraneous solutions: Since we can't take the logarithm of a negative number, $x = -2$ is an extraneous solution. Therefore, the only valid solution is $x = 8$ .

Conclusion

Great job, guys! We've successfully tackled the logarithmic equation $\log _2(9 x)-\log _2 3=3$ and worked through some practice problems. Remember, understanding the properties of logarithms and practicing consistently are key to mastering these types of equations. Keep practicing, and you'll become a pro at solving logarithmic equations in no time! If you have any questions or want to explore more complex problems, feel free to ask. Happy solving!