Welcome to SciMarvels! In this video, we solve a challenging math Olympiad problem involving exponential equations to find the value of x. 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
• Exponential Equations
• Algebraic Manipulations
• Problem-Solving Strategies
• Logarithmic Functions
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✅ Detailed Steps
We begin with the problem: If 8^x + 2^x = 130, find x.
1. Express 8 in terms of its prime factors: 8 = 2^3 (2^3)^x + 2^x = 130
2. Simplify the exponents: 2^(3x) + 2^x = 130
3. Introduce a substitution for simplification: Let t = 2^x
4. Rewrite the equation in terms of t: t^3 + t = 130
5. Factor the equation: t^3 + t = 130 t^3 + t - 130 = 0
6. Recognize that t must be a factor of 130: Factors of 130: 1, 2, 5, 10, 13, 26, 65, 130 Test values for t:
o If t = 5, then 5^3 + 5 = 130
o 125 + 5 = 130 (This works)
7. Substitute back to solve for x: t = 5 2^x = 5
8. Take the logarithm of both sides to solve for x: log(2^x) = log(5)
9. Apply the power rule of logarithms: x * log(2) = log(5)
10. Solve for x: x = log(5) / log(2)
Therefore, the value of x is: x = log(5) / log(2)
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✅ Learning Outcomes
By the end of this video, you will:
• Understand how to manipulate and solve exponential equations.
• Learn to use logarithms to solve for variables in exponentiation.
• Gain skills in algebraic manipulation and problem-solving strategies.
• Develop a deeper understanding of how to approach and solve complex math problems.
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✅ Also you can see:
• How to Solve Math Olym...
• How to Solve Math Olym...
• How to solve 7^x + 7^x...
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✅ Why Subscribe to SciMarvels?
At SciMarvels, we are dedicated to making complex math and physics problems accessible and enjoyable. Our unique approach of solving problems without voiceovers allows you to concentrate solely on the math, fostering a deeper understanding. Subscribe to join a community of learners who are passionate about mastering scientific concepts through clear, step-by-step solutions.
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