Step-by-step solution of a logarithmic equation leading to quadratic form and final solutions.

Let’s solve the equation step by step:

We are given:
log₂(x) + log₂(x - 3) = 2

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Step 1: Use the logarithm property to combine logs

When you add two logs with the same base, you can multiply their insides:

> logₐ(A) + logₐ(B) = logₐ(A × B)

So we rewrite the left side:

→ log₂[ x × (x - 3) ] = 2

Which simplifies to:

→ log₂(x² - 3x) = 2

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Step 2: Convert from log form to exponential form

Remember:
> logₐ(B) = C means a^C = B

So here, base is 2, exponent is 2, so:

→ 2² = x² - 3x

→ 4 = x² - 3x

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Step 3: Rearrange into a quadratic equation

Move everything to one side:

→ x² - 3x - 4 = 0

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Step 4: Factor the quadratic

We need two numbers that multiply to -4 and add to -3.

Those numbers are -4 and +1.

So:

→ (x - 4)(x + 1) = 0

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Step 5: Solve for x

Set each factor equal to zero:

→ x - 4 = 0 → x = 4
→ x + 1 = 0 → x = -1

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Step 6: Check for valid solutions (IMPORTANT!)

Logarithms are only defined for positive numbers. So we must check if our answers make the original logs valid.

Original equation: log₂(x) + log₂(x - 3) = 2

Check x = 4:

→ log₂(4) is OK (4 > 0)
→ log₂(4 - 3) = log₂(1) is OK (1 > 0)
✔ Valid solution.

Check x = -1:

→ log₂(-1) → NOT ALLOWED! Log of negative number is undefined.
→ Also, log₂(-1 - 3) = log₂(-4) → also undefined.
✘ Invalid solution.

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Final Answer:

x = 4