log(2)(x-1)<=1 inequation

Given the inequality:
$$\left(x - 1\right) \log{\left(2 \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(x - 1\right) \log{\left(2 \right)} = 1$$
Solve:
Given the equation:

log(2)*(x-1) = 1

Expand expressions:

-log(2) + x*log(2) = 1

Reducing, you get:

-1 - log(2) + x*log(2) = 0

Expand brackets in the left part

-1 - log2 + x*log2 = 0

Move free summands (without x)
from left part to right part, we given:
$$x \log{\left(2 \right)} - \log{\left(2 \right)} = 1$$
Divide both parts of the equation by (-log(2) + x*log(2))/x

x = 1 / ((-log(2) + x*log(2))/x)

We get the answer: x = 1 + 1/log(2)
$$x_{1} = 1 + \frac{1}{\log{\left(2 \right)}}$$
$$x_{1} = 1 + \frac{1}{\log{\left(2 \right)}}$$
This roots
$$x_{1} = 1 + \frac{1}{\log{\left(2 \right)}}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(1 + \frac{1}{\log{\left(2 \right)}}\right)$$
=
$$\frac{9}{10} + \frac{1}{\log{\left(2 \right)}}$$
substitute to the expression
$$\left(x - 1\right) \log{\left(2 \right)} \leq 1$$
$$\left(-1 + \left(\frac{9}{10} + \frac{1}{\log{\left(2 \right)}}\right)\right) \log{\left(2 \right)} \leq 1$$

/  1      1   \            
|- -- + ------|*log(2) <= 1
\  10   log(2)/            

the solution of our inequality is:
$$x \leq 1 + \frac{1}{\log{\left(2 \right)}}$$

 _____          
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       x1