Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation exp(x)=0
-
Equation x^3-3x^2=0
-
Equation 0=2-3x
-
Equation 3x-7=x-11
- Express {x} in terms of y where:
- -17*x-12*y=-4
- 9*x-19*y=-20
- 15*x+17*y=-10
- -4*x+14*y=-4
-
Identical expressions
- exp(x*(n+m))=H*n*m
- exponent of (x multiply by (n plus m)) equally H multiply by n multiply by m
- exp(x(n+m))=Hnm
- expxn+m=Hnm
-
Similar expressions
- exp(x*(n-m))=H*n*m
exp(x*(n+m))=H*n*m equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$e^{x \left(m + n\right)} = m h n$$
or
$$- m h n + e^{x \left(m + n\right)} = 0$$
or
$$\left(e^{m + n}\right)^{x} = h m n$$
or
$$\left(e^{m + n}\right)^{x} = h m n$$
- this is the simplest exponential equation
Do replacement
$$v = \left(e^{m + n}\right)^{x}$$
we get
$$- h m n + v = 0$$
or
$$- h m n + v = 0$$
Divide both parts of the equation by (v - h*m*n)/v
We get the answer: v = h*m*n
do backward replacement
$$\left(e^{m + n}\right)^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(e^{m + n} \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(h m n \right)}}{\log{\left(e^{m + n} \right)}} = \frac{\log{\left(h m n \right)}}{\log{\left(e^{m + n} \right)}}$$
$$e^{x \left(m + n\right)} = m h n$$
or
$$- m h n + e^{x \left(m + n\right)} = 0$$
or
$$\left(e^{m + n}\right)^{x} = h m n$$
or
$$\left(e^{m + n}\right)^{x} = h m n$$
- this is the simplest exponential equation
Do replacement
$$v = \left(e^{m + n}\right)^{x}$$
we get
$$- h m n + v = 0$$
or
$$- h m n + v = 0$$
Divide both parts of the equation by (v - h*m*n)/v
v = 0 / ((v - h*m*n)/v)
We get the answer: v = h*m*n
do backward replacement
$$\left(e^{m + n}\right)^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(e^{m + n} \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(h m n \right)}}{\log{\left(e^{m + n} \right)}} = \frac{\log{\left(h m n \right)}}{\log{\left(e^{m + n} \right)}}$$
The graph
Rapid solution
[src]
/log(h*m*n)\ /log(h*m*n)\
x1 = I*im|----------| + re|----------|
\ m + n / \ m + n /
$$x_{1} = \operatorname{re}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)} + i \operatorname{im}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)}$$
x1 = re(log(h*m*n)/(m + n)) + i*im(log(h*m*n)/(m + n))
Sum and product of roots
[src]
sum
/log(h*m*n)\ /log(h*m*n)\
I*im|----------| + re|----------|
\ m + n / \ m + n /
$$\operatorname{re}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)} + i \operatorname{im}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)}$$
=
/log(h*m*n)\ /log(h*m*n)\
I*im|----------| + re|----------|
\ m + n / \ m + n /
$$\operatorname{re}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)} + i \operatorname{im}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)}$$
product
/log(h*m*n)\ /log(h*m*n)\
I*im|----------| + re|----------|
\ m + n / \ m + n /
$$\operatorname{re}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)} + i \operatorname{im}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)}$$
=
/log(h*m*n)\ /log(h*m*n)\
I*im|----------| + re|----------|
\ m + n / \ m + n /
$$\operatorname{re}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)} + i \operatorname{im}{\left(\frac{\log{\left(h m n \right)}}{m + n}\right)}$$
i*im(log(h*m*n)/(m + n)) + re(log(h*m*n)/(m + n))