Solve for x: log_5(x + 1) - log_5(x - 1) = 1.

₹0.0

Question: Solve for x: log_5(x + 1) - log_5(x - 1) = 1.

Options:

Correct Answer: 2

Solution:

Using properties of logarithms: log_5((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 5 => x + 1 = 5(x - 1) => 4x = 6 => x = 2.

Solve for x: log_5(x + 1) - log_5(x - 1) = 1.

Solve for x: log_5(x + 1) - log_5(x - 1) = 1.

Correct Answer: 2

Step 1: Start with the equation: log_5(x + 1) - log_5(x - 1) = 1.
Step 2: Use the property of logarithms that states log_a(b) - log_a(c) = log_a(b/c). Apply this to the left side: log_5((x + 1)/(x - 1)) = 1.
Step 3: Rewrite the equation in exponential form. Since log_5(y) = 1 means y = 5, we have (x + 1)/(x - 1) = 5.
Step 4: Cross-multiply to eliminate the fraction: x + 1 = 5(x - 1).
Step 5: Distribute the 5 on the right side: x + 1 = 5x - 5.
Step 6: Rearrange the equation to isolate x. Subtract x from both sides: 1 = 5x - x - 5, which simplifies to 1 = 4x - 5.
Step 7: Add 5 to both sides: 1 + 5 = 4x, so 6 = 4x.
Step 8: Divide both sides by 4 to solve for x: x = 6/4, which simplifies to x = 2.

Logarithmic Properties – Understanding and applying the properties of logarithms, particularly the quotient rule.
Algebraic Manipulation – Solving linear equations after transforming logarithmic expressions.
Domain Restrictions – Recognizing the restrictions on the variable x due to the logarithmic function.