Question

Решите неравенство lg ^2 100x- 5 lg x > 6.

Answer 1

Решение во вложении......

Answer 2

To solve the inequality lg^2(100x) - 5lg(x) > 6, we'll first simplify the expressions using logarithm properties and then solve for x.

Here are the steps:

Step 1: Simplify lg^2(100x) to 2 * lg(100x) since lg^2(a) = 2 * lg(a): 2 * lg(100x) - 5lg(x) > 6

Step 2: Combine the logarithmic terms using logarithm rules: lg(100x)^2 - 5lg(x) > 6

Step 3: Express lg(100x)^2 as (lg(100x))^2, and simplify further: (lg(100x))^2 - 5lg(x) > 6

Step 4: Let's substitute lg(100x) with a variable, say a: a^2 - 5lg(x) > 6

Step 5: Move the constant term to the other side of the inequality: a^2 - 5lg(x) - 6 > 0

Step 6: Factor the quadratic expression: (a - 6)(a + 1) > 0

Now, we have a compound inequality in terms of a. To find the values of a, we'll solve the following two inequalities:

a - 6 > 0 (1) a + 1 > 0 (2)

Solving (1): a > 6

Solving (2): a > -1

Step 7: Now, recall that a = lg(100x). So, we have two inequalities for a:

lg(100x) > 6 (3) lg(100x) > -1 (4)

Step 8: We need to solve these two inequalities separately:

Solving (3): lg(100x) > 6

Converting to exponential form: 100x > 10^6

Solving for x: x > 10^6 / 100 x > 10000

Solving (4): lg(100x) > -1

Converting to exponential form: 100x > 10^(-1)

Solving for x: x > 10^(-1) / 100 x > 0.01

Step 9: Combining the results: x > 10000 and x > 0.01

Since x must satisfy both inequalities, the final solution is x > 10000.

Therefore, the solution to the original inequality is x > 10000.