If log_2(x) + log_2(x - 3) = 3, what is the value of x?
Practice Questions
Q1
If log_2(x) + log_2(x - 3) = 3, what is the value of x?
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Questions & Step-by-Step Solutions
If log_2(x) + log_2(x - 3) = 3, what is the value of x?
Correct Answer: 6
Step 1: Start with the equation log_2(x) + log_2(x - 3) = 3.
Step 2: Use the property of logarithms that says log_a(b) + log_a(c) = log_a(b * c). So, combine the logs: log_2(x(x - 3)) = 3.
Step 3: Rewrite the equation in exponential form. This means that if log_2(something) = 3, then something = 2^3. So, we have x(x - 3) = 8.
Step 4: Expand the left side: x^2 - 3x = 8.
Step 5: Rearrange the equation to set it to zero: x^2 - 3x - 8 = 0.
Step 6: Now, solve the quadratic equation x^2 - 3x - 8 = 0 using the quadratic formula or factoring.
Step 7: The solutions to the equation give us the possible values for x. In this case, we find that x = 6.
Logarithmic Properties – The question tests the understanding of properties of logarithms, specifically the product rule which states that the sum of logarithms is the logarithm of the product.
Quadratic Equations – The solution involves solving a quadratic equation derived from the logarithmic equation, testing the ability to factor or use the quadratic formula.
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