If log base 2 of x and log base 2 of y are distinct positive integers and log base x of 2 plus log base y of 2 equal to 0.5. xy=?

log_2(x) = 1/log_x(2), so

If log_2(x) = a and log_2(y) = b, we have

1/a + 1/b = 1/2
1/3 + 1/6 = 1/2

so, x=2^3 and y=2^6 so xy = 512

Well, well, well, looks like we have some logarithmic fun here! Let's break it down.

We know that log base 2 of x and log base 2 of y are distinct positive integers. So, let's say log base 2 of x = a and log base 2 of y = b, where a ≠ b.

Now, we're given that log base x of 2 + log base y of 2 = 0.5. But wait, that doesn't look entertaining enough. Let's spice it up with some ridiculous humor!

Imagine a bear holding a sign that says, "Hey, logs! No diving in the pool!" And these logs, x and y, are like, "But we wanna dive!" So, they work together, log base x of 2 and log base y of 2, and they meet halfway, which is 0.5. They are determined to make that splash, you know!

Now, let's make a splash in solving this equation. We can rewrite log base x of 2 + log base y of 2 as log base x of 2 multiplied by log base y of 2. So:

log base x of 2 * log base y of 2 = 0.5

But since log base 2 of x = a and log base 2 of y = b, we can rewrite this equation as:

a * b = 0.5

Now, let's bring the punchline! We want to find the value of xy, so let's solve for it by taking x to the power of a and y to the power of b:

xy = 2^a * 2^b

Since 2^a = x and 2^b = y, we can substitute these values in:

xy = x * y

Oh, well, would you look at that! We're right back where we started. Turns out, the value of xy is simply xy. So, there you have it!

The value of xy is just the product of x and y. Keep those logs out of the pool, my friend!

Let's solve this step by step.

Step 1: Given that log base 2 of x and log base 2 of y are distinct positive integers.

Let's assume that log base 2 of x is equal to m, and log base 2 of y is equal to n, where m and n are distinct positive integers.

So, we can write:

log base 2 of x = m

log base 2 of y = n

Step 2: Using the properties of logarithms, we can rewrite the second equation as:

log base x of 2 + log base y of 2 = 0.5

Using the logarithmic property log base b of a = 1 / (log base a of b), we can rewrite the equation as:

1 / (log base 2 of x) + 1 / (log base 2 of y) = 0.5

Substituting the values of m and n from Step 1, we get:

1 / m + 1 / n = 0.5

Step 3: Now, let's solve for m and n.

Since m and n are distinct positive integers, let's try some values:

Let's try m = 1 and n = 2:

1 / 1 + 1 / 2 = 1 + 1/2 = 1.5

Let's try m = 2 and n = 3:

1 / 2 + 1 / 3 = 0.5 + 0.33 ≈ 0.83

Let's try m = 3 and n = 4:

1 / 3 + 1 / 4 = 0.33 + 0.25 = 0.58

After testing a few values, we can see that the only solution that satisfies the equation 1 / m + 1 / n = 0.5 is m = 2 and n = 3.

Step 4: Now, let's find the values of x and y.

Since log base 2 of x = 2, we can rewrite this as:

2 = log base 2 of x

2^2 = x

4 = x

Similarly, since log base 2 of y = 3, we can rewrite this as:

3 = log base 2 of y

2^3 = y

8 = y

Step 5: Finally, let's find the value of xy.

xy = 4 * 8 = 32

So, xy = 32.

Therefore, the value of xy is 32.

To solve this problem, we need to follow a step-by-step approach.

Let's start by understanding the given information:
- log base 2 of x and log base 2 of y are distinct positive integers.
- log base x of 2 plus log base y of 2 equals 0.5.

Step 1: Find the values of log base 2 of x and log base 2 of y.
Since log base 2 of x and log base 2 of y are distinct positive integers, we can assign values to them.
Let's assume log base 2 of x = m and log base 2 of y = n, where m and n are distinct positive integers.

Step 2: Use the properties of logarithms to rewrite the second equation.
log base x of 2 plus log base y of 2 equals 0.5 can be rewritten as:
1/log base 2 of x + 1/log base 2 of y = 0.5.

Step 3: Apply the change of base formula to further simplify the equation.
Using the change of base formula, we can rewrite the equation as:
log base x of 2 + log base y of 2 = log base 2 of 2 * 0.5.

Step 4: Simplify the right side of the equation.
log base 2 of 2 is equal to 1. Therefore, we get:
log base x of 2 + log base y of 2 = 1 * 0.5,
log base x of 2 + log base y of 2 = 0.5.

Step 5: Use properties of logarithms to combine the two logarithmic terms.
Apply the logarithmic rule: log base a of b + log base a of c = log base a of (b * c).
We can rewrite the equation as:
log base xy of 2 = 0.5.

Step 6: Exponentiate both sides of the equation using the base xy.
By exponentiating both sides, we will have:
2 = (xy)^0.5.

Step 7: Square both sides of the equation to solve for xy.
2^2 = (xy)^0.5^2,
4 = xy.

Therefore, xy is equal to 4.

In conclusion, the value of xy is 4 based on the given information and the steps we followed to solve the problem.