Question

Решить логарифмическое неравенство:

Answer 1

log0.5 (x+3) <log0.25 (x+15)

log0.25 (x+3)^2 <log0.25 (x+15)

x^2+6x+9<x+15

x^2+5x-6<0

Решим нер-во x^2+5x-6=0

Д=25-4*1*(-6)=49

x1=(-5+7)/2=1

x2=(-5-7)/2=-6

х принадлежит (-6;1)

Answer 2

Solving the Logarithmic Inequality

To solve the logarithmic inequality \(log_2(x-4)=log_{4x}4x\), we can follow these steps:

1. Start by setting the bases of the logarithms equal to each other: \(log_2(x-4)=log_22\). This is because \(log_{4x}4x\) simplifies to \(log_22\) since the base and the argument are the same.

2. Since \(log_2(x-4)=log_22\) simplifies to \(log_2(x-4)=1\), we can equate the exponents: \(x-4=2\).

3. Solve for \(x\): \(x=6\).

4. Check the validity of the solution. Since the original inequality is \(log_2(x-4)=log_{4x}4x\), we need to ensure that \(x-4\) and \(4x\) are both positive.

- For \(x=6\), \(x-4=6-4=2\) and \(4x=4\cdot6=24\). Both \(x-4\) and \(4x\) are positive, so the solution \(x=6\) is valid.

Therefore, the solution to the logarithmic inequality \(log_2(x-4)=log_{4x}4x\) is \(x=6\).