Question

Log0,5 (2x)>2 Решите уравнение

Answer 1

{x>0
{2x<0,25⇒x<0,125
x∈(0;0,125)

Answer 2

To solve the inequality $\log_{0.5}(2x) > 2$, we need to follow these steps:

Step 1: Rewrite the inequality in exponential form. Step 2: Solve for $x$ in the exponential inequality. Step 3: Check for any extraneous solutions.

Let's proceed with each step:

Step 1: Rewrite the inequality in exponential form. In exponential form, $\log_{a}(b) = c$ is equivalent to $a^c = b$. So, for our inequality $\log_{0.5}(2x) > 2$, we rewrite it as: $0.5^2 > 2x$

Step 2: Solve for $x$ in the exponential inequality. $0.5^2 > 2x$ $0.25 > 2x$

Now, divide both sides by 2 to isolate $x$: $0.25/2 > x$ $0.125 > x$

Step 3: Check for any extraneous solutions. The domain of the logarithm is $x > 0$ since we can't take the logarithm of a non-positive number. Therefore, the solution $x = 0.125$ is valid.

So, the solution to the inequality $\log_{0.5}(2x) > 2$ is $x > 0.125$.