Mister Exam
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- Equation:
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Equation x/9+x=-10/3
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Equation √(x^2-1)=2*x
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Equation 4(x-3)=x+6
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Identical expressions
- (one / two)*ln(two *y+ one)=C-ln(cos(x))
- (1 divide by 2) multiply by ln(2 multiply by y plus 1) equally C minus ln( co sinus of e of (x))
- (one divide by two) multiply by ln(two multiply by y plus one) equally C minus ln( co sinus of e of (x))
- (1/2)ln(2y+1)=C-ln(cos(x))
- 1/2ln2y+1=C-lncosx
- (1 divide by 2)*ln(2*y+1)=C-ln(cos(x))
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Similar expressions
- (1/2)*ln(2*y+1)=C+ln(cos(x))
- (1/2)*ln(2*y-1)=C-ln(cos(x))
- (1/2)*ln(2*y+1)=C-ln(cosx)
(1/2)*ln(2*y+1)=C-ln(cos(x)) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
log(2*y + 1)
------------ = c - log(cos(x))
2
$$\frac{\log{\left(2 y + 1 \right)}}{2} = c - \log{\left(\cos{\left(x \right)} \right)}$$
Detail solution
Given the equation
$$\frac{\log{\left(2 y + 1 \right)}}{2} = c - \log{\left(\cos{\left(x \right)} \right)}$$
$$\frac{\log{\left(2 y + 1 \right)}}{2} = c - \log{\left(\cos{\left(x \right)} \right)}$$
Let's divide both parts of the equation by the multiplier of log =1/2
$$\log{\left(2 y + 1 \right)} = 2 c - 2 \log{\left(\cos{\left(x \right)} \right)}$$
This equation is of the form:
By definition log
then
simplify
$$2 y + 1 = \frac{e^{2 c}}{\cos^{2}{\left(x \right)}}$$
$$2 y = \frac{e^{2 c}}{\cos^{2}{\left(x \right)}} - 1$$
$$y = \frac{e^{2 c}}{2 \cos^{2}{\left(x \right)}} - \frac{1}{2}$$
$$\frac{\log{\left(2 y + 1 \right)}}{2} = c - \log{\left(\cos{\left(x \right)} \right)}$$
$$\frac{\log{\left(2 y + 1 \right)}}{2} = c - \log{\left(\cos{\left(x \right)} \right)}$$
Let's divide both parts of the equation by the multiplier of log =1/2
$$\log{\left(2 y + 1 \right)} = 2 c - 2 \log{\left(\cos{\left(x \right)} \right)}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
c - log(cos(x))
---------------
1/2
2*y + 1 = e simplify
$$2 y + 1 = \frac{e^{2 c}}{\cos^{2}{\left(x \right)}}$$
$$2 y = \frac{e^{2 c}}{\cos^{2}{\left(x \right)}} - 1$$
$$y = \frac{e^{2 c}}{2 \cos^{2}{\left(x \right)}} - \frac{1}{2}$$
The graph
Sum and product of roots
[src]
sum
/ 2*c \ / 2*c \
| e | | e |
re|-------| I*im|-------|
| 2 | | 2 |
1 \cos (x)/ \cos (x)/
- - + ----------- + -------------
2 2 2
$$\frac{\operatorname{re}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} + \frac{i \operatorname{im}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} - \frac{1}{2}$$
=
/ 2*c \ / 2*c \
| e | | e |
re|-------| I*im|-------|
| 2 | | 2 |
1 \cos (x)/ \cos (x)/
- - + ----------- + -------------
2 2 2
$$\frac{\operatorname{re}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} + \frac{i \operatorname{im}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} - \frac{1}{2}$$
product
/ 2*c \ / 2*c \
| e | | e |
re|-------| I*im|-------|
| 2 | | 2 |
1 \cos (x)/ \cos (x)/
- - + ----------- + -------------
2 2 2
$$\frac{\operatorname{re}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} + \frac{i \operatorname{im}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} - \frac{1}{2}$$
=
/ 2*c \ / 2*c \
| e | | e |
re|-------| I*im|-------|
| 2 | | 2 |
1 \cos (x)/ \cos (x)/
- - + ----------- + -------------
2 2 2
$$\frac{\operatorname{re}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} + \frac{i \operatorname{im}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} - \frac{1}{2}$$
-1/2 + re(exp(2*c)/cos(x)^2)/2 + i*im(exp(2*c)/cos(x)^2)/2
Rapid solution
[src]
/ 2*c \ / 2*c \
| e | | e |
re|-------| I*im|-------|
| 2 | | 2 |
1 \cos (x)/ \cos (x)/
y1 = - - + ----------- + -------------
2 2 2
$$y_{1} = \frac{\operatorname{re}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} + \frac{i \operatorname{im}{\left(\frac{e^{2 c}}{\cos^{2}{\left(x \right)}}\right)}}{2} - \frac{1}{2}$$
y1 = re(exp(2*c)/cos(x)^2)/2 + i*im(exp(2*c)/cos(x)^2)/2 - 1/2