Solve Log₂x + Log₂(x/2) = 5: Finding the Value of x

To solve this equation, we follow these steps:

• Step 1: Use the property of logarithms $\log_b a + \log_b c = \log_b (a \cdot c)$ to combine terms on the left-hand side of the equation.
• Step 2: Simplify the expression under the logarithm and solve for $x$.

Let's proceed through these steps:

Step 1: Rewrite the equation using logarithmic properties:
$\log_2 x + \log_2 \frac{x}{2} = \log_2 x + \log_2 x - \log_2 2$

This simplifies to:
$2 \log_2 x - 1 = 5$

Step 2: Solve the equation:
Add 1 to both sides:

$2 \log_2 x = 6$

Divide both sides by 2:

$\log_2 x = 3$

Now, convert the logarithmic equation to its exponential form:

$x = 2^3$

Calculate $x$:

$x = 8$

Therefore, the solution to the problem is $x = 8$.