Solving The Logarithmic Equation: Log₄(x) + Log₁₆(x) - 2log₆₄(x) = 5/2

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Hey guys! Today, we're diving deep into the world of logarithms to solve a pretty interesting equation: log₄(x) + log₁₆(x) - 2log₆₄(x) = 5/2. Logarithmic equations might seem daunting at first, but with a few key properties and a step-by-step approach, we can crack this nut wide open. So, buckle up, grab your thinking caps, and let's get started!

Understanding Logarithms

Before we jump into the solution, let's quickly recap what logarithms are all about. In simple terms, a logarithm answers the question: "To what power must we raise the base to get a certain number?" For instance, log₂ 8 = 3 because 2 raised to the power of 3 equals 8 (2³ = 8). The general form is logₐ(b) = c, which means aᶜ = b, where 'a' is the base, 'b' is the argument (the number we're taking the logarithm of), and 'c' is the exponent (the logarithm itself).

Key Properties of Logarithms

To effectively solve logarithmic equations, we need to be familiar with some crucial properties. Here are a few that we'll use in our solution:

  • Change of Base Formula: This allows us to convert logarithms from one base to another. It states that logₐ(b) = logₓ(b) / logₓ(a), where 'x' can be any base (often 10 or e for calculators).
  • Power Rule: This rule tells us that logₐ(bⁿ) = n * logₐ(b). In other words, if we have an exponent inside the logarithm, we can bring it out as a coefficient.
  • Logarithm of a Product: logₐ(b * c) = logₐ(b) + logₐ(c). This property allows us to break down the logarithm of a product into the sum of individual logarithms.
  • Logarithm of a Quotient: logₐ(b / c) = logₐ(b) - logₐ(c). Similar to the product rule, this one helps us deal with division inside logarithms.

With these properties in our arsenal, we're well-equipped to tackle the given equation.

Step-by-Step Solution

Now, let's get our hands dirty and solve the equation log₄(x) + log₁₆(x) - 2log₆₄(x) = 5/2 step by step. Our main goal here is to simplify the equation by expressing all logarithms in the same base. This will allow us to combine them and eventually isolate 'x'.

1. Change All Logarithms to Base 2

Looking at the bases 4, 16, and 64, we can see that they are all powers of 2 (4 = 2², 16 = 2⁴, and 64 = 2⁶). This makes base 2 a convenient choice for our common base. We'll use the change of base formula to convert each term:

  • log₄(x) = log₂(x) / log₂(4) = log₂(x) / 2
  • log₁₆(x) = log₂(x) / log₂(16) = log₂(x) / 4
  • log₆₄(x) = log₂(x) / log₂(64) = log₂(x) / 6

Now, let's substitute these back into our original equation:

(log₂(x) / 2) + (log₂(x) / 4) - 2(log₂(x) / 6) = 5/2

2. Simplify the Equation

Next, we'll simplify the equation by getting rid of the fractions. Notice that 2(log₂(x) / 6) simplifies to (log₂(x) / 3). So our equation now looks like this:

(log₂(x) / 2) + (log₂(x) / 4) - (log₂(x) / 3) = 5/2

To combine these terms, we need a common denominator. The least common multiple of 2, 4, and 3 is 12. Multiply each term to get the common denominator:

(6log₂(x) / 12) + (3log₂(x) / 12) - (4log₂(x) / 12) = 5/2

Now we can combine the numerators:

(6log₂(x) + 3log₂(x) - 4log₂(x)) / 12 = 5/2

Simplifying the numerator gives us:

5log₂(x) / 12 = 5/2

3. Isolate log₂(x)

To isolate log₂(x), we'll multiply both sides of the equation by 12/5:

log₂(x) = (5/2) * (12/5)

log₂(x) = 6

4. Solve for x

Now we have a simple logarithmic equation. To solve for 'x', we'll rewrite the equation in exponential form. Remember that logₐ(b) = c is equivalent to aᶜ = b. So:

log₂(x) = 6 means 2⁶ = x

Calculating 2⁶ gives us:

x = 64

So, the solution to our equation is x = 64.

Verification

It's always a good idea to verify our solution by plugging it back into the original equation. Let's do that:

log₄(64) + log₁₆(64) - 2log₆₄(64) = 5/2

We know that:

  • log₄(64) = 3 (since 4³ = 64)
  • log₁₆(64) = 3/2 (since 16^(3/2) = (16^(1/2))³ = 4³ = 64)
  • log₆₄(64) = 1 (since 64¹ = 64)

Substituting these values, we get:

3 + 3/2 - 2(1) = 5/2

3 + 3/2 - 2 = 5/2

(6/2) + (3/2) - (4/2) = 5/2

5/2 = 5/2

Our solution checks out! This gives us confidence that our answer is correct.

Common Mistakes to Avoid

When working with logarithmic equations, there are a few common pitfalls to watch out for:

  • Forgetting the Change of Base Formula: If you have logarithms with different bases, you'll need to use the change of base formula to express them in a common base.
  • Incorrectly Applying Logarithmic Properties: Make sure you're using the properties of logarithms correctly. For example, logₐ(b + c) is NOT equal to logₐ(b) + logₐ(c).
  • Ignoring the Domain of Logarithms: Remember that the argument of a logarithm must be positive. Always check your solutions to make sure they don't result in taking the logarithm of a non-positive number.
  • Algebraic Errors: Simple algebraic mistakes can throw off your entire solution. Double-check each step to ensure accuracy.

Practice Makes Perfect

Like any mathematical skill, solving logarithmic equations becomes easier with practice. Try working through a variety of problems, and don't be afraid to make mistakes – they're a great way to learn! If you get stuck, revisit the fundamental properties of logarithms and try breaking the problem down into smaller steps.

Conclusion

So there you have it! We've successfully solved the equation log₄(x) + log₁₆(x) - 2log₆₄(x) = 5/2. By using the properties of logarithms and following a systematic approach, we were able to simplify the equation and find the solution x = 64. Remember, the key to mastering logarithms is understanding their properties and practicing regularly.

Keep practicing, and you'll become a logarithm whiz in no time! If you have any questions or want to explore more challenging problems, feel free to ask. Happy solving!