Question

Решить неравенство log₀,₅≤1. log₀,₄>1

Answer 1

1) log₀,₅х≤1
log₀,₅х≤log₀,₅0,5
х≥0,5

ОДЗ: х>0
Объединив решение с ОДЗ: х≥0,5
Ответ: [0,5; +∞)

2) log₀,₄х>1
log₀,₄х>log₀,₄0,4
х<0,4

ОДЗ: х>0
Объединив решение с ОДЗ: 0<х<0,4
Ответ: (0;0,4]

Answer 2

To solve the inequalities, we need to first understand the properties of logarithms. The logarithm of a number represents the power to which a given base must be raised to obtain that number. Let's solve the inequalities step by step:

1. log₀,₅ ≤ 1: We want to find the values of x for which the logarithm of 0.5 (base 0.5) is less than or equal to 1.

The logarithm is a decreasing function, so log(a) ≤ log(b) if and only if a ≥ b.

Therefore, log₀,₅ ≤ 1 is equivalent to 0.5 ≥ 0.5¹.

Since 0.5¹ is just 0.5 itself, we have: 0.5 ≥ 0.5

This statement is true for all values of x. So the solution to log₀,₅ ≤ 1 is the entire domain of real numbers.

2. log₀,₄ > 1: Here, we want to find the values of x for which the logarithm of 0.4 (base 0.4) is greater than 1.

Again, using the property of logarithms, we have:

log₀,₄ > 1 is equivalent to 0.4 > 4¹.

Since 4¹ is just 4 itself, we have: 0.4 > 4

This statement is not true for any values of x. So, there are no solutions to log₀,₄ > 1.

In summary:

1. The solution to log₀,₅ ≤ 1 is (-∞, +∞) (the entire domain of real numbers).
2. There are no solutions to log₀,₄ > 1.