Solving Logarithmic Equations: Log₄(3x) - Log₄(9) = 1

Let's dive into solving a logarithmic equation! Logarithmic equations might seem intimidating at first, but with a step-by-step approach and understanding of logarithm properties, they can be tackled with confidence. In this article, we will solve the equation log₄(3x) - log₄(9) = 1. We'll break down each step, explain the underlying principles, and provide insights to help you grasp the concepts thoroughly. Whether you're a student learning logarithms or just brushing up on your math skills, this guide is designed to be clear, comprehensive, and helpful.

Understanding Logarithms

Before we start solving the equation, let's briefly revisit what logarithms are and some key properties that will be useful.

A logarithm is essentially the inverse operation to exponentiation. If we have an equation like bˣ = y, we can express it in logarithmic form as log_b(y) = x. Here, 'b' is the base of the logarithm, 'x' is the exponent, and 'y' is the result of the exponentiation. Understanding this relationship is fundamental to manipulating and solving logarithmic equations.

Several properties of logarithms are particularly useful for simplifying and solving equations. Here are a few:

1. Product Rule: log_b(mn) = log_b(m) + log_b(n)
2. Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
3. Power Rule: log_b(m^p) = p * log_b(m)
4. Change of Base Formula: log_b(a) = log_c(a) / log_c(b)

For the equation we're tackling today, the quotient rule will be particularly handy. Now, let's apply these concepts to solve the given equation.

Solving the Equation log₄(3x) - log₄(9) = 1

Our equation is log₄(3x) - log₄(9) = 1. The first step is to use the quotient rule to combine the two logarithmic terms on the left side of the equation. Recall that the quotient rule states log_b(m) - log_b(n) = log_b(m/n). Applying this to our equation, we get:

log₄((3x)/9) = 1

Simplify the fraction inside the logarithm:

log₄(x/3) = 1

Now that we have a single logarithmic term, we can convert the equation from logarithmic form to exponential form. Remember that log_b(y) = x is equivalent to bˣ = y. In our case, b = 4, x = 1, and y = x/3. So, we can rewrite the equation as:

4¹ = x/3

This simplifies to:

4 = x/3

To solve for x, multiply both sides of the equation by 3:

3 * 4 = x

12 = x

So, the solution to the equation log₄(3x) - log₄(9) = 1 is x = 12. It's always a good idea to check your solution by plugging it back into the original equation to make sure it holds true.

Verification

Let's verify our solution by substituting x = 12 back into the original equation:

log₄(3 * 12) - log₄(9) = 1

log₄(36) - log₄(9) = 1

Using the quotient rule again:

log₄(36/9) = 1

log₄(4) = 1

Since 4¹ = 4, the equation holds true. Therefore, our solution x = 12 is correct.

Common Mistakes to Avoid

When working with logarithmic equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them:

1. Incorrectly Applying Logarithmic Properties: Make sure to apply the product, quotient, and power rules correctly. A common mistake is to misapply these rules, especially when dealing with more complex expressions.
2. Forgetting the Base: Always remember the base of the logarithm. The properties and rules change depending on the base. In our example, the base was 4, which is crucial for converting between logarithmic and exponential forms.
3. Not Checking the Solution: Always verify your solution by plugging it back into the original equation. This is especially important because logarithmic functions have domain restrictions. The argument of a logarithm must be positive. If your solution makes the argument of any logarithm negative or zero, it is not a valid solution.
4. Ignoring Domain Restrictions: Logarithmic functions are only defined for positive arguments. Before solving, identify any restrictions on x imposed by the logarithms in the equation. For instance, in the equation log₄(3x) - log₄(9) = 1, 3x must be greater than 0, which means x > 0. If you find a solution that does not satisfy these restrictions, it must be discarded.
5. Assuming Log(A - B) = Log(A) - Log(B): This is a frequent error. There isn't a direct simplification for log(A - B). Remember, the quotient rule applies to log(A/B), which is log(A) - log(B).

Additional Practice Problems

To solidify your understanding of solving logarithmic equations, here are a few practice problems:

1. Solve: log₂(x + 3) + log₂(x - 3) = 4
2. Solve: log₅(2x - 1) = 2
3. Solve: 2log₃(x) = log₃(16)

Try solving these problems on your own, and then check your answers. Working through these exercises will help you build confidence and proficiency in solving logarithmic equations.

Conclusion

Solving logarithmic equations involves understanding the properties of logarithms, converting between logarithmic and exponential forms, and carefully applying these concepts step by step. By avoiding common mistakes and practicing regularly, you can become proficient in solving a wide range of logarithmic equations. In this article, we successfully solved the equation log₄(3x) - log₄(9) = 1, verified the solution, and discussed common pitfalls to avoid. Keep practicing, and you'll master the art of solving logarithmic equations in no time!

For further reading and a deeper dive into logarithmic functions, you might find valuable resources on Khan Academy's Algebra II section, which offers lessons, practice exercises, and videos on various topics, including logarithms.