Question

4. log_2(x - 3) + 3 = log_2(3x + 1)

Answer 1

Ответ: log_2(x - 3) + 3 = log_2(3x + 1)

x=5

Answer 2

To solve the equation log₂(x - 3) + 3 = log₂(3x + 1), you can use the properties of logarithms. Here are the steps to solve it:

1. Start by subtracting log₂(x - 3) from both sides of the equation to isolate the logarithmic term on one side:

   3 = log₂(3x + 1) - log₂(x - 3)

2. Use the property of logarithms that states that the difference of logarithms with the same base is equal to the logarithm of the quotient:

   3 = log₂[(3x + 1) / (x - 3)]

3. Now, remove the logarithm by converting the equation into an exponential form:

   2^3 = (3x + 1) / (x - 3)

4. Simplify the left side:

   8 = (3x + 1) / (x - 3)

5. To get rid of the fraction, multiply both sides of the equation by (x - 3):

   8(x - 3) = 3x + 1

6. Distribute on the left side:

   8x - 24 = 3x + 1

7. Move the 3x term to the left side by subtracting 3x from both sides:

   8x - 3x - 24 = 1

8. Simplify:

   5x - 24 = 1

9. Add 24 to both sides:

   5x = 1 + 24 5x = 25

10. Finally, divide both sides by 5 to isolate x:

x = 25 / 5 x = 5

So, the solution to the equation log₂(x - 3) + 3 = log₂(3x + 1) is x = 5.