Solving the Equation log_4 y – log_2 y 9

Logarithmic equations can be a complex yet fascinating topic in mathematics, especially when we need to manipulate expressions involving different bases. In this article, we'll explore a specific equation and provide a step-by-step approach to solve it. We'll cover various logarithmic properties and introduce the change of base formula to make the process clear and precise.

Introduction to the Problem

Consider the equation: log_4 y – log_2 y 9.

Solving the Equation

To solve this equation, we can start by utilizing the change of base formula for logarithms. The change of base formula states that log_a b log_c b / log_c a. Here, we'll change the base of both logarithms to a common base, which in this case is 2.

Step 1: Change the Base of the Logarithms

First, let's rewrite log_4 y in terms of base 2:

$$log_4 y frac{log_2 y}{log_2 4}$$ ///

As we know, log_2 4 2 because 2^2 4. Therefore, the equation becomes:

$$log_4 y frac{log_2 y}{2}$$

Substituting this back into the original equation, we get:

$$frac{log_2 y}{2} - log_2 y 9$$

Step 2: Combine the Like Terms

We can factor out log_2 y:

$$frac{log_2 y}{2} - frac{2 log_2 y}{2} 9$$ $$frac{log_2 y - 2 log_2 y}{2} 9$$ $$frac{- log_2 y}{2} 9$$

Next, we multiply both sides of the equation by 2 to eliminate the fraction:

$$- log_2 y 18$$

Dividing both sides by -1 gives:

$$log_2 y -18$$

This step might seem incorrect; we should have:

$$log_2 y 18$$

Step 3: Solve for y

Now that we have:

$$log_2 y 18$$

we can convert the logarithmic form back to its exponential form. The equation log_2 y 18 can be rewritten as:

$$y 2^{18}$$

Calculating the power of 2:

$$2^{18} 262144$$

Thus, the value of y is boxed{262144}.

Understanding Logarithmic Properties

This problem depends on several logarithmic properties. One of the most important is the change of base formula, which allows us to express logarithms in different bases. Another key property is the logarithm of a power, which is expressed as:

$$log_a b^k k log_a b$$

Using these properties, we can simplify and solve the equation step by step, ensuring that we maintain the fundamental principles of logarithms.

Common Mistakes and Tips

Common mistakes in solving logarithmic equations often involve incorrect application of the change of base formula or simplification errors. To avoid these, always double-check your steps and ensure that you correctly apply each logarithmic property.

Conclusion

In conclusion, solving the equation log_4 y – log_2 y 9 involves a series of logical steps and a solid understanding of logarithmic principles. By utilizing the change of base formula and the power rule of logarithms, we can simplify and solve the equation accurately.