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Solve for x: log_3(x + 1) - log_3(x - 1) = 1.
Solve for x: log_3(x + 1) - log_3(x - 1) = 1.
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Solve for x: log_3(x + 1) - log_3(x - 1) = 1.
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Using properties of logarithms, log_3((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 3 => x + 1 = 3(x - 1) => x = 2.
Questions & Step-by-step Solutions
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Q: Solve for x: log_3(x + 1) - log_3(x - 1) = 1.
Solution:
Using properties of logarithms, log_3((x + 1)/(x - 1)) = 1 => (x + 1)/(x - 1) = 3 => x + 1 = 3(x - 1) => x = 2.
Steps: 8
Show Steps
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.
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