Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- log1/2*log5(x-2)>0
- lg(2x-7)>0
- cos(2x+(pi/6))<=-sqrt(3)/2
- cosx/(1+cos2x)<0
- Integral of d{x}:
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1,5
- Sum of series:
-
1,5
-
Identical expressions
- log three (x)-logx(3)< one , five
- logarithm of 3(x) minus logarithm of x(3) less than 1,5
- logarithm of three (x) minus logarithm of x(3) less than one , five
- log3x-logx3<1,5
-
Similar expressions
- log3(x)+logx(3)<1,5
log3(x)-logx(3)<1,5 inequation
The solution
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[src]
log(x) ------ - log(x)*3 < 3/2 log(3)
$$- 3 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} < \frac{3}{2}$$
-3*log(x) + log(x)/log(3) < 3/2
Detail solution
Given the inequality:
$$- 3 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} < \frac{3}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$- 3 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} = \frac{3}{2}$$
Solve:
$$x_{1} = 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
$$x_{1} = 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
This roots
$$x_{1} = 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
=
$$- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
substitute to the expression
$$- 3 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} < \frac{3}{2}$$
$$\frac{\log{\left(- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(3 \right)}} - 3 \log{\left(- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}} \right)} < \frac{3}{2}$$
but
Then
$$x < 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
no execute
the solution of our inequality is:
$$x > 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
$$- 3 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} < \frac{3}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$- 3 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} = \frac{3}{2}$$
Solve:
$$x_{1} = 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
$$x_{1} = 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
This roots
$$x_{1} = 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
=
$$- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
substitute to the expression
$$- 3 \log{\left(x \right)} + \frac{\log{\left(x \right)}}{\log{\left(3 \right)}} < \frac{3}{2}$$
$$\frac{\log{\left(- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}} \right)}}{\log{\left(3 \right)}} - 3 \log{\left(- \frac{1}{10} + 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}} \right)} < \frac{3}{2}$$
/ 3 \
| ----------------|
/ 3 \ | 1 2*(1 - 3*log(3))|
| ----------------| log|- -- + 3 | < 3/2
| 1 2*(1 - 3*log(3))| \ 10 /
- 3*log|- -- + 3 | + -----------------------------
\ 10 / log(3) but
/ 3 \
| ----------------|
/ 3 \ | 1 2*(1 - 3*log(3))|
| ----------------| log|- -- + 3 | > 3/2
| 1 2*(1 - 3*log(3))| \ 10 /
- 3*log|- -- + 3 | + -----------------------------
\ 10 / log(3) Then
$$x < 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
no execute
the solution of our inequality is:
$$x > 3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}$$
_____
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Solving inequality on a graph
Rapid solution
[src]
/ -3 \ | ----------------- | | 2*(-1 + 3*log(3)) | And\3 < x, x < oo/
$$3^{- \frac{3}{2 \left(-1 + 3 \log{\left(3 \right)}\right)}} < x \wedge x < \infty$$
(x < oo)∧(3^(-3/(2*(-1 + 3*log(3)))) < x)
Rapid solution 2
[src]
3 ---------------- 2*(1 - 3*log(3)) (3 , oo)
$$x\ in\ \left(3^{\frac{3}{2 \left(1 - 3 \log{\left(3 \right)}\right)}}, \infty\right)$$
x in Interval.open(3^(3/(2*(1 - 3*log(3)))), oo)