Question

(0.3)^log5(log1/5(x^2-4/5)<1

Answer 1

Ты шо такие сложные задачи решаешь

Answer 2

Let's solve the inequality step by step. First, rewrite the inequality as follows:

(0.3)^(log5(log(1/5(x^2-4/5))) < 1

Now, we can simplify the expression inside the logarithm:

log(1/5(x^2-4/5)) = log(1/5) + log(x^2-4/5)

Next, apply the logarithmic properties:

log(1/5) is a constant, let's call it "C".

So, the inequality becomes:

(0.3)^(log5(C + log(x^2-4/5))) < 1

Now, focus on the exponent of 0.3:

log5(C + log(x^2-4/5))

Since the base of the logarithm is 5, let's try to rewrite the equation using the change of base formula:

log_a(b) = log_c(b) / log_c(a)

Here, "a" is the base of the logarithm, "b" is the argument of the logarithm, and "c" is the new base.

Apply the change of base formula to the exponent:

log5(C + log(x^2-4/5)) = log(C + log(x^2-4/5)) / log(5)

Now the inequality becomes:

(0.3)^(log(C + log(x^2-4/5)) / log(5)) < 1

To simplify it further, consider the following property of exponentiation:

a^(b/c) = (a^b)^(1/c)

Apply this to our inequality:

[(0.3)^log(C + log(x^2-4/5))]^(1/log(5)) < 1

Now, we have:

0.3^(log(C + log(x^2-4/5))) < 1

Since the base 0.3 is between 0 and 1, raising it to any positive exponent will result in a number less than 1. Therefore, this inequality holds for any value of "x."

So, the solution to the inequality is all real numbers: (-∞, +∞).