Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 2x-3=10
-
Equation 4*x^4-5*x^2+1=0
-
Equation 5*x^2+7*x+2=0
-
Equation 3/8*x+15=1/6*x+10
- Express {x} in terms of y where:
- -2*x+4*y=3
- -14*x-13*y=11
- -1*x-5*y=-1
- -12*x-6*y=-3
-
Identical expressions
- three ^x- two * three ^x- two = seven
- 3 to the power of x minus 2 multiply by 3 to the power of x minus 2 equally 7
- three to the power of x minus two multiply by three to the power of x minus two equally seven
- 3x-2*3x-2=7
- 3^x-23^x-2=7
- 3x-23x-2=7
-
Similar expressions
- 3^x-2*3^x+2=7
- 3^x+2*3^x-2=7
3^x-2*3^x-2=7 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(- 2 \cdot 3^{x} + 3^{x}\right) - 2 = 7$$
or
$$\left(\left(- 2 \cdot 3^{x} + 3^{x}\right) - 2\right) - 7 = 0$$
or
$$- 3^{x} = 9$$
or
$$3^{x} = -9$$
- this is the simplest exponential equation
Do replacement
$$v = 3^{x}$$
we get
$$v + 9 = 0$$
or
$$v + 9 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = -9$$
We get the answer: v = -9
do backward replacement
$$3^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(3 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(-9 \right)}}{\log{\left(3 \right)}} = \frac{\log{\left(9 \right)} + i \pi}{\log{\left(3 \right)}}$$
$$\left(- 2 \cdot 3^{x} + 3^{x}\right) - 2 = 7$$
or
$$\left(\left(- 2 \cdot 3^{x} + 3^{x}\right) - 2\right) - 7 = 0$$
or
$$- 3^{x} = 9$$
or
$$3^{x} = -9$$
- this is the simplest exponential equation
Do replacement
$$v = 3^{x}$$
we get
$$v + 9 = 0$$
or
$$v + 9 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = -9$$
We get the answer: v = -9
do backward replacement
$$3^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(3 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(-9 \right)}}{\log{\left(3 \right)}} = \frac{\log{\left(9 \right)} + i \pi}{\log{\left(3 \right)}}$$
Sum and product of roots
[src]
sum
log(9) pi*I ------ + ------ log(3) log(3)
$$\frac{\log{\left(9 \right)}}{\log{\left(3 \right)}} + \frac{i \pi}{\log{\left(3 \right)}}$$
=
log(9) pi*I ------ + ------ log(3) log(3)
$$\frac{\log{\left(9 \right)}}{\log{\left(3 \right)}} + \frac{i \pi}{\log{\left(3 \right)}}$$
product
log(9) pi*I ------ + ------ log(3) log(3)
$$\frac{\log{\left(9 \right)}}{\log{\left(3 \right)}} + \frac{i \pi}{\log{\left(3 \right)}}$$
=
pi*I + log(9)
-------------
log(3)
$$\frac{\log{\left(9 \right)} + i \pi}{\log{\left(3 \right)}}$$
(pi*i + log(9))/log(3)
Rapid solution
[src]
log(9) pi*I
x1 = ------ + ------
log(3) log(3)
$$x_{1} = \frac{\log{\left(9 \right)}}{\log{\left(3 \right)}} + \frac{i \pi}{\log{\left(3 \right)}}$$
x1 = log(9)/log(3) + i*pi/log(3)