✅ Welcome to SciMarvels! In this video, we solve a challenging Iranian math Olympiad problem involving logarithmic equations. Watch as we solve the problem step-by-step on paper, providing a clear and detailed explanation without any voiceover. This unique approach allows you to focus entirely on the math and understand the process visually.
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✅ Key Concepts Covered
• Logarithmic Equations
• Algebraic Manipulations
• Problem-Solving Strategies
• Logarithmic Properties
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✅ Detailed Steps
We begin with the problem: If log_2(x^2 + 2x + 4) + log_2(x - 2) = 3, find log_(∛2) (x).
1. Combine the logarithmic expressions using the product rule:
Log_2((x - 2)(x^2 + 2x + 4)) = 3
2. Set the argument of the logarithm equal to the base raised to the power on the right-hand side:
(x - 2)(x^2 + 2x + 4) = 2^3
(x - 2)(x^2 + 2x + 4) = 8
3. Expand the left-hand side:
x^3 + 2x^2 + 4x - 2x^2 - 4x - 8 = 8
4. Simplify the equation:
x^3 - 8 = 8
x^3 = 16
5. Solve for x:
x = ∛(16)
x = 2^(4/3)
6. Substitute x into log_(∛2) (x):
log_(∛2) (2^(4/3))
7. Apply the change of base formula:
log_(∛2) (2^(4/3)) = (4/3) * log_(∛2) 2
8. Express log_(∛2) 2 using properties of logarithms:
log_(∛2) 2 = log2 2 / log2 (∛2)
log_(∛2) 2 = 1 / (log2 2^(1/3))
log_(∛2) 2 = 1 / (1/3 * log2 2)
log_(∛2) 2 = 1 / (1/3)
log_(∛2) 2 = 3
9. Combine the results:
log_(∛2) (2^(4/3)) = (4/3) * 3
log_(∛2) (2^(4/3)) = 4
Therefore, the value of log_(∛2) x is: log_(∛2) x = 4
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• How to Solve Exponenti...
• How to Solve Math Olym...
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✅ Learning Outcomes
By the end of this video, you will:
• Understand how to solve complex logarithmic equations.
• Learn to apply logarithmic properties and algebraic manipulations.
• Gain skills in problem-solving strategies for math Olympiad problems.
• Develop a deeper understanding of how to approach and solve advanced math problems.
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