Question

Log1/7^(x-8)≤log1/7(13-x)

Answer 1

{x-8>0⇒x>8
{13-x>0⇒x<13
{x-8≥13-x⇒x+x≥13+8⇒2x≥21⇒x≥10,5
x∈[10,5;13)

Answer 2

To solve this inequality involving logarithmic expressions, we can use the properties of logarithms to simplify and analyze the inequality. The properties we'll use are:

1. If a = b, then log₁₇(a) = log₁₇(b).
2. If a > 1, then logₐ(x) is an increasing function.
3. If a < 1, then logₐ(x) is a decreasing function.

Given inequality:

log₁/₇^(x-8) ≤ log₁/₇(13-x)

We can simplify the logarithmic expressions using property 1:

x - 8 ≤ 13 - x

Now, let's solve for x:

Add x to both sides:

2x - 8 ≤ 13

Add 8 to both sides:

2x ≤ 21

Divide both sides by 2:

x ≤ 10.5

So, the solution for the inequality is:

x ≤ 10.5