Question

Log1/2(7x^2-200)=log1/2 50x

Answer 1

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Answer 2

To solve the equation log₁/₂(7x² - 200) = log₁/₂(50x), you can use the properties of logarithms.

First, notice that both sides have the same base, which is 1/2, so you can set the expressions inside the logarithms equal to each other:

7x² - 200 = 50x

Now, let's move all terms to one side of the equation to set it to zero:

7x² - 50x - 200 = 0

Now, you have a quadratic equation. To solve it, you can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this equation, a = 7, b = -50, and c = -200. Plug these values into the quadratic formula:

x = (-(-50) ± √((-50)² - 4(7)(-200))) / (2(7))

Simplify:

x = (50 ± √(2500 + 5600)) / 14

x = (50 ± √8100) / 14

x = (50 ± 90) / 14

Now, you have two possible solutions:

1. x = (50 + 90) / 14 = 140 / 14 = 10
2. x = (50 - 90) / 14 = -40 / 14 = -20/7

So, the solutions to the equation log₁/₂(7x² - 200) = log₁/₂(50x) are x = 10 and x = -20/7.