Mister Exam
log3(13-4^x)>2 inequation
The solution
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/ x\ log\13 - 4 / ------------ > 2 log(3)
$$\frac{\log{\left(13 - 4^{x} \right)}}{\log{\left(3 \right)}} > 2$$
log(13 - 4^x)/log(3) > 2
Detail solution
Given the inequality:
$$\frac{\log{\left(13 - 4^{x} \right)}}{\log{\left(3 \right)}} > 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(13 - 4^{x} \right)}}{\log{\left(3 \right)}} = 2$$
Solve:
$$x_{1} = 1$$
$$x_{1} = 1$$
This roots
$$x_{1} = 1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1$$
=
$$\frac{9}{10}$$
substitute to the expression
$$\frac{\log{\left(13 - 4^{x} \right)}}{\log{\left(3 \right)}} > 2$$
$$\frac{\log{\left(13 - 4^{\frac{9}{10}} \right)}}{\log{\left(3 \right)}} > 2$$
the solution of our inequality is:
$$x < 1$$
$$\frac{\log{\left(13 - 4^{x} \right)}}{\log{\left(3 \right)}} > 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{\log{\left(13 - 4^{x} \right)}}{\log{\left(3 \right)}} = 2$$
Solve:
$$x_{1} = 1$$
$$x_{1} = 1$$
This roots
$$x_{1} = 1$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 1$$
=
$$\frac{9}{10}$$
substitute to the expression
$$\frac{\log{\left(13 - 4^{x} \right)}}{\log{\left(3 \right)}} > 2$$
$$\frac{\log{\left(13 - 4^{\frac{9}{10}} \right)}}{\log{\left(3 \right)}} > 2$$
/ 4/5\
log\13 - 2*2 /
---------------- > 2
log(3)
the solution of our inequality is:
$$x < 1$$
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Solving inequality on a graph