Mister Exam
log2(x-1)<2 inequation
The solution
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log(x - 1) ---------- < 2 log(2)
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} < 2$$
log(x - 1*1)/log(2) < 2
Detail solution
Given the inequality:
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = 2$$
Solve:
Given the equation
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = 2$$
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = 2$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(x - 1 \right)} = 2 \log{\left(2 \right)}$$
This equation is of the form:
By definition log
then
$$1 x - 1 = e^{\frac{2}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$x - 1 = 4$$
$$x = 5$$
$$x_{1} = 5$$
$$x_{1} = 5$$
This roots
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$\frac{49}{10}$$
substitute to the expression
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} < 2$$
$$\frac{\log{\left(\left(-1\right) 1 + \frac{49}{10} \right)}}{\log{\left(2 \right)}} < 2$$
the solution of our inequality is:
$$x < 5$$
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = 2$$
Solve:
Given the equation
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = 2$$
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} = 2$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(x - 1 \right)} = 2 \log{\left(2 \right)}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$1 x - 1 = e^{\frac{2}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$x - 1 = 4$$
$$x = 5$$
$$x_{1} = 5$$
$$x_{1} = 5$$
This roots
$$x_{1} = 5$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$\frac{49}{10}$$
substitute to the expression
$$\frac{\log{\left(x - 1 \right)}}{\log{\left(2 \right)}} < 2$$
$$\frac{\log{\left(\left(-1\right) 1 + \frac{49}{10} \right)}}{\log{\left(2 \right)}} < 2$$
/39\ log|--| \10/ < 2 ------- log(2)
the solution of our inequality is:
$$x < 5$$
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x_1
Solving inequality on a graph