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

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

The correct answer and explanation is:

To solve the equation: log⁡2(3x−7)=3\log_2(3x – 7) = 3

Step-by-Step Solution:

Step 1: Understand what the equation means.

This is a logarithmic equation with base 2. It says: log⁡2(3x−7)=3\log_2(3x – 7) = 3

This means: 3x−7=233x – 7 = 2^3

Step 2: Calculate 232^3

23=82^3 = 8

Now the equation becomes: 3x−7=83x – 7 = 8

Step 3: Solve for xx

Add 7 to both sides: 3x=153x = 15

Now divide both sides by 3: x=5x = 5

Final Answer:

5\boxed{5}

Explanation (Approx. 300 Words):

This equation involves a logarithm with base 2. A logarithmic equation such as log⁡b(A)=C\log_b(A) = C is equivalent to an exponential form: A=bCA = b^C. This is the key idea for solving logarithmic equations.

In the given equation, log⁡2(3x−7)=3\log_2(3x – 7) = 3, the expression inside the logarithm is 3x−73x – 7, the base is 2, and the result is 3. Rewriting it in exponential form gives us: 3x−7=233x – 7 = 2^3

Calculating 232^3 gives 8, so we simplify: 3x−7=83x – 7 = 8

This is a basic linear equation. We solve it by first isolating 3x3x through addition: 3x=153x = 15

Then, divide both sides by 3: x=5x = 5

However, when solving logarithmic equations, it’s essential to check if the solution is valid. Logarithms are only defined for positive arguments. That means: 3x−7>0⇒x>733x – 7 > 0 \Rightarrow x > \frac{7}{3}

Our solution, x=5x = 5, satisfies this condition, so it is valid.

Therefore, the only correct solution is: 5\boxed{5}