Mister Exam
Other calculators
-
How to use it?
- Equation:
- Equation 5*x^2-45=0
-
Equation x^3+4*x^2-9*x-36=0
-
Equation 6-5(4x-1)=3
-
Equation x^2+4x-21=0
- Express {x} in terms of y where:
- -3*x-19*y=-9
- -4*x-12*y=13
- -19*x+10*y=-3
- 20*x-9*y=-8
- Sum of series:
-
81
-
Identical expressions
- three ^(four *x- two)= eighty-one
- 3 to the power of (4 multiply by x minus 2) equally 81
- three to the power of (four multiply by x minus two) equally eighty minus one
- 3(4*x-2)=81
- 34*x-2=81
- 3^(4x-2)=81
- 3(4x-2)=81
- 34x-2=81
- 3^4x-2=81
-
Similar expressions
- 680,35*10^(-9)=(2*61,8*10^(-9))/(3*10^8)*sqrt((2*1,38*10^(-23)*x)/0,048)
- 3^(4*x+2)=81
- 1*2*3+2*4*6+4*8*12+7*14*21/1*3*5+2*6*10+4*12*20+7*21*35=×
3^(4*x-2)=81 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$3^{4 x - 2} = 81$$
or
$$3^{4 x - 2} - 81 = 0$$
or
$$\frac{81^{x}}{9} = 81$$
or
$$81^{x} = 729$$
- this is the simplest exponential equation
Do replacement
$$v = 81^{x}$$
we get
$$v - 729 = 0$$
or
$$v - 729 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 729$$
We get the answer: v = 729
do backward replacement
$$81^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(81 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(729 \right)}}{\log{\left(81 \right)}} = \frac{3}{2}$$
$$3^{4 x - 2} = 81$$
or
$$3^{4 x - 2} - 81 = 0$$
or
$$\frac{81^{x}}{9} = 81$$
or
$$81^{x} = 729$$
- this is the simplest exponential equation
Do replacement
$$v = 81^{x}$$
we get
$$v - 729 = 0$$
or
$$v - 729 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 729$$
We get the answer: v = 729
do backward replacement
$$81^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(81 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(729 \right)}}{\log{\left(81 \right)}} = \frac{3}{2}$$
Sum and product of roots
[src]
sum
3 log(27) pi*I log(27) pi*I 3 pi*I - + -------- - -------- + -------- + -------- + - + ------ 2 2*log(3) 2*log(3) 2*log(3) 2*log(3) 2 log(3)
$$\left(\left(\frac{3}{2} + \left(\frac{\log{\left(27 \right)}}{2 \log{\left(3 \right)}} - \frac{i \pi}{2 \log{\left(3 \right)}}\right)\right) + \left(\frac{\log{\left(27 \right)}}{2 \log{\left(3 \right)}} + \frac{i \pi}{2 \log{\left(3 \right)}}\right)\right) + \left(\frac{3}{2} + \frac{i \pi}{\log{\left(3 \right)}}\right)$$
=
log(27) pi*I
3 + ------- + ------
log(3) log(3)
$$3 + \frac{\log{\left(27 \right)}}{\log{\left(3 \right)}} + \frac{i \pi}{\log{\left(3 \right)}}$$
product
/log(27) pi*I \
3*|-------- - --------|
\2*log(3) 2*log(3)/ /log(27) pi*I \ /3 pi*I \
-----------------------*|-------- + --------|*|- + ------|
2 \2*log(3) 2*log(3)/ \2 log(3)/
$$\frac{3 \left(\frac{\log{\left(27 \right)}}{2 \log{\left(3 \right)}} - \frac{i \pi}{2 \log{\left(3 \right)}}\right)}{2} \left(\frac{\log{\left(27 \right)}}{2 \log{\left(3 \right)}} + \frac{i \pi}{2 \log{\left(3 \right)}}\right) \left(\frac{3}{2} + \frac{i \pi}{\log{\left(3 \right)}}\right)$$
=
3*(pi*I + log(27))*(-pi*I + log(27))*(2*pi*I + log(27))
-------------------------------------------------------
3
16*log (3)
$$\frac{3 \left(\log{\left(27 \right)} - i \pi\right) \left(\log{\left(27 \right)} + i \pi\right) \left(\log{\left(27 \right)} + 2 i \pi\right)}{16 \log{\left(3 \right)}^{3}}$$
3*(pi*i + log(27))*(-pi*i + log(27))*(2*pi*i + log(27))/(16*log(3)^3)
Rapid solution
[src]
x1 = 3/2
$$x_{1} = \frac{3}{2}$$
log(27) pi*I
x2 = -------- - --------
2*log(3) 2*log(3)
$$x_{2} = \frac{\log{\left(27 \right)}}{2 \log{\left(3 \right)}} - \frac{i \pi}{2 \log{\left(3 \right)}}$$
log(27) pi*I
x3 = -------- + --------
2*log(3) 2*log(3)
$$x_{3} = \frac{\log{\left(27 \right)}}{2 \log{\left(3 \right)}} + \frac{i \pi}{2 \log{\left(3 \right)}}$$
3 pi*I
x4 = - + ------
2 log(3)
$$x_{4} = \frac{3}{2} + \frac{i \pi}{\log{\left(3 \right)}}$$
x4 = 3/2 + i*pi/log(3)
Numerical answer
[src]
x1 = 1.5
x2 = 1.5 - 1.42980043369006*i
x3 = 1.5 + 1.42980043369006*i
x4 = 1.5 + 2.85960086738013*i
x4 = 1.5 + 2.85960086738013*i