Solve for x: log_3(x + 1) - log_3(x - 1) = 1.

₹0.0

Question: Solve for x: log_3(x + 1) - log_3(x - 1) = 1.

Options:

Correct Answer: 2

Solution:

Using properties of logarithms, log_3((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 3 => x + 1 = 3(x - 1) => x = 2.

Solve for x: log_3(x + 1) - log_3(x - 1) = 1.

Solve for x: log_3(x + 1) - log_3(x - 1) = 1.

Correct Answer: 2

Step 1: Start with the equation: log_3(x + 1) - log_3(x - 1) = 1.
Step 2: Use the property of logarithms that states log_a(b) - log_a(c) = log_a(b/c). So, rewrite the left side: log_3((x + 1)/(x - 1)) = 1.
Step 3: Convert the logarithmic equation to an exponential form. This means that if log_3(y) = 1, then y = 3. So, set (x + 1)/(x - 1) = 3.
Step 4: Now, solve the equation (x + 1)/(x - 1) = 3. Multiply both sides by (x - 1) to eliminate the fraction: x + 1 = 3(x - 1).
Step 5: Distribute the 3 on the right side: x + 1 = 3x - 3.
Step 6: Rearrange the equation to isolate x. Subtract x from both sides: 1 = 2x - 3.
Step 7: Add 3 to both sides: 4 = 2x.
Step 8: Divide both sides by 2 to solve for x: x = 2.

Logarithmic Properties – Understanding how to manipulate logarithmic expressions, particularly the quotient rule.
Exponential Equations – Converting logarithmic equations into exponential form to solve for the variable.
Domain Restrictions – Recognizing the restrictions on the variable due to the logarithmic function, ensuring the arguments are positive.