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If log_2(x) + log_2(x - 3) = 3, what is the value of x?
If log_2(x) + log_2(x - 3) = 3, what is the value of x?
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If log_2(x) + log_2(x - 3) = 3, what is the value of x?
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log_2(x(x - 3)) = 3 => x(x - 3) = 2^3 = 8 => x^2 - 3x - 8 = 0. Solving gives x = 6.
Questions & Step-by-step Solutions
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Q: If log_2(x) + log_2(x - 3) = 3, what is the value of x?
Solution:
log_2(x(x - 3)) = 3 => x(x - 3) = 2^3 = 8 => x^2 - 3x - 8 = 0. Solving gives x = 6.
Steps: 7
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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.
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