Solving Logarithmic Equations: Find Solutions For 2 Log₅(x) = Log₅(4)

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Hey guys! Let's dive into solving a logarithmic equation. Today, we're tackling the equation 2 log₅(x) = log₅(4). This might seem tricky at first, but don't worry, we'll break it down step by step. Understanding logarithmic equations is crucial in many areas of mathematics and even in real-world applications, so let's get started!

Understanding Logarithms

Before we jump into the solution, let's quickly recap what logarithms are all about. A logarithm is essentially the inverse operation to exponentiation. In simpler terms, if we have an equation like aᵇ = c, then the logarithm of c to the base a is b, written as logₐ(c) = b. Think of it as asking: "To what power must we raise 'a' to get 'c'?"

Key Properties of Logarithms

To solve our equation, we need to remember some key properties of logarithms:

  • Power Rule: logₐ(xⁿ) = n logₐ(x). This rule allows us to move exponents inside the logarithm out as coefficients, and vice versa.
  • Logarithm of a Product: logₐ(xy) = logₐ(x) + logₐ(y). The logarithm of a product is the sum of the logarithms.
  • Logarithm of a Quotient: logₐ(x/y) = logₐ(x) - logₐ(y). The logarithm of a quotient is the difference of the logarithms.
  • One-to-One Property: If logₐ(x) = logₐ(y), then x = y. This property is super important because it allows us to get rid of the logarithms once we have a single logarithm on each side of the equation.

Knowing these properties will be our secret weapon in solving the equation 2 log₅(x) = log₅(4). So, keep these in mind as we move forward!

Solving the Equation 2 log₅(x) = log₅(4)

Okay, let's get to the main event: solving the equation 2 log₅(x) = log₅(4). Our goal is to isolate 'x', but first, we need to deal with those logarithms. Remember our power rule? This is where it comes in handy!

Step 1: Apply the Power Rule

The power rule states that logₐ(xⁿ) = n logₐ(x). Looking at our equation, we have the term 2 log₅(x). We can use the power rule to rewrite this as log₅(x²). So, our equation now becomes:

log₅(x²) = log₅(4)

See how we've simplified the left side by moving the coefficient 2 into the exponent? This is a crucial step in making the equation easier to handle. Now, both sides of the equation have a single logarithm with the same base.

Step 2: Use the One-to-One Property

This is where the magic happens! The one-to-one property tells us that if logₐ(x) = logₐ(y), then x = y. In our case, we have log₅(x²) = log₅(4). Since the bases are the same (both are base 5), we can simply equate the arguments:

x² = 4

We've successfully eliminated the logarithms! Now we're left with a simple quadratic equation. Isn't that neat?

Step 3: Solve the Quadratic Equation

Now we need to solve x² = 4. There are a couple of ways to do this. The easiest way is to take the square root of both sides. Remember, when we take the square root, we need to consider both the positive and negative roots:

√(x²) = ±√4

This gives us:

x = ±2

So, we have two potential solutions: x = 2 and x = -2. But hold on a second! We're not quite done yet.

Checking for Extraneous Solutions

This is a super important step when dealing with logarithmic equations. Not every solution we find algebraically is actually a valid solution. Why? Because we can't take the logarithm of a negative number or zero. So, we need to check our potential solutions in the original equation to make sure they work.

Checking x = 2

Let's plug x = 2 into the original equation: 2 log₅(x) = log₅(4)

2 log₅(2) = log₅(4)

Using the power rule, we can rewrite the left side as:

log₅(2²) = log₅(4)

log₅(4) = log₅(4)

This is true! So, x = 2 is a valid solution.

Checking x = -2

Now let's plug in x = -2: 2 log₅(-2) = log₅(4)

Here's the problem: we can't take the logarithm of a negative number. log₅(-2) is undefined. Therefore, x = -2 is not a valid solution. It's what we call an extraneous solution – a solution that we found algebraically but doesn't work in the original equation.

Final Answer

After all that work, we've arrived at the final answer! The only valid solution for the equation 2 log₅(x) = log₅(4) is:

x = 2

See? Logarithmic equations aren't so scary once you break them down and remember those key properties. Always remember to check for extraneous solutions to make sure your answer is correct!

Key Takeaways

Let's quickly recap the key steps we took to solve this equation:

  1. Apply the Power Rule: Rewrite the equation to get a single logarithm on each side.
  2. Use the One-to-One Property: Equate the arguments of the logarithms.
  3. Solve the Equation: Solve the resulting algebraic equation.
  4. Check for Extraneous Solutions: Make sure your solutions are valid by plugging them back into the original equation.

By following these steps, you can tackle a wide variety of logarithmic equations with confidence. Keep practicing, and you'll become a log-solving pro in no time!

Practice Problems

Want to test your skills? Try solving these similar logarithmic equations:

  1. 3 log₂(x) = log₂(8)
  2. 2 log₃(x) = log₃(25)
  3. log₄(x²) = log₄(9)

Share your answers in the comments below! We can discuss the solutions and help each other learn.

Conclusion

So, there you have it! We've successfully solved the equation 2 log₅(x) = log₅(4) and learned some valuable strategies for tackling logarithmic equations. Remember the properties, be careful with extraneous solutions, and keep practicing. You've got this!

Understanding logarithms opens doors to many advanced mathematical concepts and real-world applications. Keep exploring, keep learning, and most importantly, have fun with math! If you found this helpful, give it a thumbs up and share it with your friends. And don't forget to subscribe for more math adventures!