Persamaan Eksponen Kelas 10 : Bentuk 2 - Matematika Peminatan || Part 2

Summary

In this video, Justin Steart Leonardo explains the second form of exponential equations for 10th-grade advanced mathematics. He begins by outlining the basic formula where if the bases are the same, the exponents can be equated, provided the base is positive and not equal to 1. Justin then solves several example problems, breaking down the process step by step, including simplifying expressions, equating exponents, and solving for variables. He concludes by encouraging viewers to try some practice problems and to like, share, and subscribe to the channel for more educational content.

Takeaways

• 📘 The video discusses exponent equations, specifically focusing on type two exponent equations in math for grade 10 specialization.
• 📐 The key formula mentioned is: if a^FX = a^GX, then FX = GX, provided that 'a' is positive and not equal to 1.
• 🧮 Example 1: The solution for 27^3x - 4 = 1/9^4 - x involves simplifying both sides to have the same base (base 3) and solving for x, yielding x = -4/5.
• 🔢 Example 2: The solution for 2^(2x + 1)√16^(x - 1) = 4^x / 8^(x + 1) involves converting everything to base 2, resulting in x = -2/5.
• 💡 Example 3: Solving 2^(x² - 3x - 5) = 1/√2^(x + 3) requires converting both sides to the same base and solving for x, yielding x = 7/2.
• 🧩 Example 4: In the equation 3^(x² + x - 4) = 9^(3x - 5), converting both sides to base 3 and solving gives x1 = 3 and x2 = 2, so the sum x1 + x2 = 5.
• ⚠️ Example 5: Solving 1/25^(x - 2.5) = √(625/5^(2 - x)) involves converting everything to base 5, resulting in x = 8/5.
• ✏️ Example 6: In √^3 from 4^(5 - x) / 8 = 1/2^(2x + 1), converting both sides to base 2 and solving results in x = -1.
• 🔍 The speaker emphasizes the importance of accuracy in solving these equations, ensuring consistent base conversions.
• 👍 The video encourages viewers to try solving four additional practice problems and promotes subscribing for more content.

Q & A

What is the basic rule for solving exponent equations with the same base?

The basic rule is that if you have the same base on both sides of the equation, you can equate the exponents. For example, if a^FX = a^GX, then FX = GX, as long as the base 'a' is positive and not equal to 1.

What is the solution for the equation 27^(3x - 4) = 1/9^(4 - x)?

First, rewrite both sides of the equation with a base of 3: 27 = 3^3 and 1/9 = 3^-2. This gives 3^(3(3x - 4)) = 3^(-2(4 - x)). By equating the exponents, you get 9x - 12 = -8 + 2x. Solving for x, you find x = -4/5.

How do you solve the equation 2^(2x + 1) * √16^(x - 1) = 4^x / 8^(x + 1)?

Start by converting all terms to base 2. √16 is rewritten as 2^(4(x - 1)/2), 4^x becomes 2^(2x), and 8^(x + 1) becomes 2^(3(x + 1)). Simplifying the exponents and equating both sides, you get 4x - 1 = -x - 3. Solving for x, you find x = -2/5.

What is the solution for the equation 2^(x² - 3x - 5) = 1/√2^(x + 3)?

Rewrite the right-hand side as 2^(-1/2(x + 3)). Equating exponents, you get x² - 3x - 5 = -1/2(x + 3). Multiplying both sides by 2 to eliminate the fraction, you solve for x using the quadratic formula. The positive solution is x = 7/2.

How do you solve for x1 + x2 in the equation 3^(x² + x - 4) = 9^(3x - 5)?

Convert 9^(3x - 5) to base 3, giving 3^(2(3x - 5)). Equate exponents: x² + x - 4 = 6x - 10. Simplifying, you get x² - 5x + 6 = 0. Factoring the quadratic equation gives x1 = 3 and x2 = 2. Therefore, x1 + x2 = 5.

How do you approach solving 1/25^(x - 2.5) = √625 / 5^(2 - x)?

Rewrite 1/25^(x - 2.5) as 5^(-2(x - 2.5)) and √625 as 5^2 / 5^(2 - x). After simplifying and equating the exponents, solve for x by moving terms to the correct sides. The solution is x = 8/5.

What steps are involved in solving √³(4^(5 - x) / 8) = 1/2^(2x + 1)?

Convert the terms to base 2. 4 becomes 2² and 8 becomes 2³. Simplifying the equation and equating the exponents, you get 10 - 2x / 3 - 3 = -2x - 1. Solving for x gives x = -1.

What is the importance of converting all terms to the same base when solving exponential equations?

Converting terms to the same base allows you to directly compare and equate the exponents, simplifying the process of solving the equation.

Why is it important that the base 'a' in exponential equations is positive and not equal to 1?

If the base 'a' is 1, any exponent will yield the same result (1), which renders the equation unsolvable in terms of finding distinct exponents. The base must also be positive to avoid complex results.

What is a common mistake to avoid when solving exponent equations involving roots?

A common mistake is forgetting to account for the square or cube root by properly adjusting the exponent. For example, √16 is equivalent to 16^(1/2), so you must remember to divide the exponent accordingly.