Mister Exam
log2(2x-4)
The solution
You have entered
[src]
log(2*x - 4) 6
------------ < log (2)
log(2)
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
log(2*x - 4)/log(2) < log(2)^6
Detail solution
Given the inequality:
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
Solve:
Given the equation
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(2 x - 4 \right)} = \log{\left(2 \right)}^{7}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$2 x - 4 = e^{\frac{\log{\left(2 \right)}^{6}}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$2 x - 4 = e^{\log{\left(2 \right)}^{7}}$$
$$2 x = e^{\log{\left(2 \right)}^{7}} + 4$$
$$x = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
This roots
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2\right)$$
=
$$\frac{e^{\log{\left(2 \right)}^{7}}}{2} + \frac{19}{10}$$
substitute to the expression
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
$$\frac{\log{\left(-4 + 2 \left(\frac{e^{\log{\left(2 \right)}^{7}}}{2} + \frac{19}{10}\right) \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
/ 7 \
| 1 log (2)|
log|- - + e | 6
\ 5 / < log (2)
-------------------
log(2)
the solution of our inequality is:
$$x < \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
_____
\
-------ο-------
x1
Solving inequality on a graph
Rapid solution
[src]
/ 7 \
| log (2)|
| e |
And|2 < x, x < 2 + --------|
\ 2 /
$$2 < x \wedge x < \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
(2 < x)∧(x < 2 + exp(log(2)^7)/2)
Rapid solution 2
[src]
7
log (2)
e
(2, 2 + --------)
2
$$x\ in\ \left(2, \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2\right)$$
x in Interval.open(2, exp(log(2)^7)/2 + 2)
The solution
You have entered
[src]
log(2*x - 4) 6 ------------ < log (2) log(2)
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
log(2*x - 4)/log(2) < log(2)^6
Detail solution
Given the inequality:
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
Solve:
Given the equation
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(2 x - 4 \right)} = \log{\left(2 \right)}^{7}$$
This equation is of the form:
By definition log
then
$$2 x - 4 = e^{\frac{\log{\left(2 \right)}^{6}}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$2 x - 4 = e^{\log{\left(2 \right)}^{7}}$$
$$2 x = e^{\log{\left(2 \right)}^{7}} + 4$$
$$x = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
This roots
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2\right)$$
=
$$\frac{e^{\log{\left(2 \right)}^{7}}}{2} + \frac{19}{10}$$
substitute to the expression
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
$$\frac{\log{\left(-4 + 2 \left(\frac{e^{\log{\left(2 \right)}^{7}}}{2} + \frac{19}{10}\right) \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
the solution of our inequality is:
$$x < \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
Solve:
Given the equation
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} = \log{\left(2 \right)}^{6}$$
Let's divide both parts of the equation by the multiplier of log =1/log(2)
$$\log{\left(2 x - 4 \right)} = \log{\left(2 \right)}^{7}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$2 x - 4 = e^{\frac{\log{\left(2 \right)}^{6}}{\frac{1}{\log{\left(2 \right)}}}}$$
simplify
$$2 x - 4 = e^{\log{\left(2 \right)}^{7}}$$
$$2 x = e^{\log{\left(2 \right)}^{7}} + 4$$
$$x = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
This roots
$$x_{1} = \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \left(\frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2\right)$$
=
$$\frac{e^{\log{\left(2 \right)}^{7}}}{2} + \frac{19}{10}$$
substitute to the expression
$$\frac{\log{\left(2 x - 4 \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
$$\frac{\log{\left(-4 + 2 \left(\frac{e^{\log{\left(2 \right)}^{7}}}{2} + \frac{19}{10}\right) \right)}}{\log{\left(2 \right)}} < \log{\left(2 \right)}^{6}$$
/ 7 \
| 1 log (2)|
log|- - + e | 6
\ 5 / < log (2)
-------------------
log(2)
the solution of our inequality is:
$$x < \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
_____
\
-------ο-------
x1
Solving inequality on a graph
Rapid solution
[src]
/ 7 \ | log (2)| | e | And|2 < x, x < 2 + --------| \ 2 /
$$2 < x \wedge x < \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2$$
(2 < x)∧(x < 2 + exp(log(2)^7)/2)
Rapid solution 2
[src]
7
log (2)
e
(2, 2 + --------)
2
$$x\ in\ \left(2, \frac{e^{\log{\left(2 \right)}^{7}}}{2} + 2\right)$$
x in Interval.open(2, exp(log(2)^7)/2 + 2)