What is the solution of log2 (3x – 7) = 3

What is the solution of log2 (3x – 7) = 3?

The correct answer and explanation is:

✅ Correct Answer:

The solution to the equation log₂(3x − 7) = 3 is:
x = 5

🧠 Step-by-Step Explanation:

We are given the logarithmic equation: log⁡2(3x−7)=3\log_2(3x – 7) = 3

This means:
“The power to which 2 must be raised to get (3x − 7) is 3.”

🔁 Step 1: Convert the logarithmic equation to exponential form

log⁡b(A)=C⇒A=bC\log_b(A) = C \quad \Rightarrow \quad A = b^C

So: 3x−7=233x – 7 = 2^3 3x−7=83x – 7 = 8

✏️ Step 2: Solve the linear equation

3x=8+7=153x = 8 + 7 = 15 x=153=5x = \frac{15}{3} = 5

✅ Step 3: Check the solution

Substitute x=5x = 5 into the original equation: log⁡2(3(5)−7)=log⁡2(15−7)=log⁡2(8)\log_2(3(5) – 7) = \log_2(15 – 7) = \log_2(8) log⁡2(8)=3(because 23=8)\log_2(8) = 3 \quad \text{(because } 2^3 = 8\text{)}

✅ This confirms the solution is correct.

⚠️ Important Note on Domain:

For any logarithmic expression like log⁡b(A)\log_b(A), the argument A must be positive.

In our case, 3x−7>03x – 7 > 0 3x>7⇒x>733x > 7 \Rightarrow x > \frac{7}{3}

Since x=5x = 5 is greater than 73\frac{7}{3}, it lies within the valid domain.

📌 Final Answer:

x=5\boxed{x = 5}

This means when you plug 5 into the expression, the logarithmic identity is satisfied.