Question

Log1/3_log0,2_log32_(x+1)/(x+7)<0

Answer 1

$log_{ \frac{1}{3} }log_{0,2}log_{32} \frac{x+1}{x+7}\ \textless \ 0$

1/3 < 1, поэтому меняем знак:

$log_{0,2}log_{32} \frac{x+1}{x+7} \ \textgreater \ ( \frac{1}{3} )^0 = 1$

0,2 < 1, поэтому меняем знак:

$log_{32} \frac{x+1}{x+7} \ \textless \ (0,2 )^1 = 0,2$

32 > 1 поэтому знак остается:

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Answer 2

To solve the inequality:

log₁/₃(log₀.₂(log₃₂((x+1)/(x+7))))) < 0

Let's break it down step by step:

1. Innermost logarithm: log₃₂((x+1)/(x+7))

   The base of this logarithm is 32, so we can rewrite it in exponential form: 32^y = (x+1)/(x+7)

2. Second logarithm: log₀.₂(log₃₂((x+1)/(x+7)))

   The base of this logarithm is 0.2, so we can rewrite it in exponential form: (0.2)^z = log₃₂((x+1)/(x+7))

3. Outer logarithm: log₁/₃(log₀.₂(log₃₂((x+1)/(x+7))))

   The base of this logarithm is 1/3, so we can rewrite it in exponential form: (1/3)^w = log₀.₂(log₃₂((x+1)/(x+7)))

Now we have:

(1/3)^w = (0.2)^z

We need to solve this equation first. Taking the logarithm of both sides will help:

w log(1/3) = z log(0.2)

Now, let's solve for z in terms of w:

z = (w log(1/3)) / log(0.2)

Next, we need to analyze the inequality:

(1/3)^w < 0

Since (1/3) is positive, we need the exponent w to be negative. Therefore:

w < 0

Finally, substituting the expression for z into the inequality:

(0.2)^[(w log(1/3)) / log(0.2)] < 0

Now, solving this inequality is not possible because raising a positive number (0.2) to any power will always be positive. Therefore, there is no solution to the given inequality.

Please note that the steps above assume that the expression is correctly written and there are no typos or errors in the original equation. Double-check the expression to ensure accuracy.