Question

Log²5(25-x²)-3log5(25-x²)+2≥0

Answer 1

$log_5^2\left ( 25-x^2 \right )-3log_5\left ( 25-x^2 \right )+2\geq 0\\\left ( log_5\left ( 25-x^2 \right )-1 \right )\left ( log_5\left ( 25-x^2 \right )-2 \right )\geq 0\\log_5\left ( 25-x^2 \right )\in \left ( -\infty ;1 \right ]\cup \left [ 2;+\infty \right )\\x\in \left ( -\infty ;-2\sqrt{5} \right ]\cup \left \{ 0 \right \}\cup \left [ 2\sqrt{5};+\infty \right )\\25-x^2>0\Rightarrow x\in \left ( -5 ;-2\sqrt{5} \right ]\cup \left \{ 0 \right \}\cup \left [ 2\sqrt{5};+5 \right )$

Answer 2

To solve the inequality log²5(25-x²) - 3log5(25-x²) + 2 ≥ 0, we can simplify the expression and solve for the value of x.

Let's start by simplifying the logarithmic terms. We'll use the properties of logarithms to rewrite the equation:

log²5(25-x²) - 3log5(25-x²) + 2 ≥ 0

Now, let's simplify each logarithmic term:

log₂5(25-x²) - log₅(25-x²)³ + 2 ≥ 0

Using the properties of logarithms, we can rewrite the equation further:

log₂5(25-x²) - log₅((25-x²)³) + 2 ≥ 0

Now, we can simplify the expression inside the logarithms:

log₂5(25-x²) - log₅((25-x²)³) + 2 ≥ 0

Next, we can apply the change of base formula to convert the logarithms to a common base:

log(25-x²)/log(2) - log((25-x²)³)/log(5) + 2 ≥ 0

Now, let's simplify further:

[log(25-x²)/log(2)] - [3log(25-x²)/log(5)] + 2 ≥ 0

Now, we can find a common denominator:

[log(25-x²)*log(5) - 3log(25-x²)log(2) + 2log(2)*log(5)] / [log(2)*log(5)] ≥ 0

Next, let's simplify the numerator:

[log(5)(log(25-x²) - 3log(25-x²)log(2) + 2log(2))] / [log(2)*log(5)] ≥ 0

Simplifying further:

[log(5)(log(25-x²) - 3log(25-x²)log(2) + 2log(2))] / [log(2)*log(5)] ≥ 0

Now, let's cancel out the common factors:

[log(25-x²) - 3log(25-x²)log(2) + 2log(2)] / [log(2)] ≥ 0

Now, let's simplify the expression inside the numerator:

[log(25-x²) - log((25-x²)³)log(2) + 2log(2)] / [log(2)] ≥ 0

Next, let's simplify the logarithmic terms inside the numerator:

[log(25-x²) - 3log(25-x²)log(2) + 2log(2)] / [log(2)] ≥ 0

Now, let's simplify the expression further:

[log(25-x²) - 3log(25-x²)log(2) + 2log(2)] / [log(2)] ≥ 0

Now, we have a complex expression involving logarithms. Unfortunately, it's not possible to solve this inequality algebraically. However, you can approximate the solution by using numerical methods or a graphing calculator.