Mister Exam
log_2(x-4)<1 inequation
The solution
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log(x - 4) ---------- < 1 log(2)
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} < 1$$
log(x - 4)/log(2) < 1
Detail solution
Given the inequality:
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} = 1$$
Solve:
Given the equation
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} = 1$$
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} = 1$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(x - 4 \right)} = \log{\left(2 \right)}$$
This equation is of the form:
By definition log
then
$$x - 4 = e^{\frac{1}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$x - 4 = 2$$
$$x = 6$$
$$x_{1} = 6$$
$$x_{1} = 6$$
This roots
$$x_{1} = 6$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 6$$
=
$$\frac{59}{10}$$
substitute to the expression
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} < 1$$
$$\frac{\log{\left(-4 + \frac{59}{10} \right)}}{\log{\left(2 \right)}} < 1$$
the solution of our inequality is:
$$x < 6$$
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} = 1$$
Solve:
Given the equation
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} = 1$$
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} = 1$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(x - 4 \right)} = \log{\left(2 \right)}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$x - 4 = e^{\frac{1}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$x - 4 = 2$$
$$x = 6$$
$$x_{1} = 6$$
$$x_{1} = 6$$
This roots
$$x_{1} = 6$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 6$$
=
$$\frac{59}{10}$$
substitute to the expression
$$\frac{\log{\left(x - 4 \right)}}{\log{\left(2 \right)}} < 1$$
$$\frac{\log{\left(-4 + \frac{59}{10} \right)}}{\log{\left(2 \right)}} < 1$$
/19\ log|--| \10/ < 1 ------- log(2)
the solution of our inequality is:
$$x < 6$$
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