Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (x+2)^4+(x+2)^2-12=0
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Equation 10x^2=80x
- Equation 4sin3x+sin5x-2sinxcos2x=0
- Equation (1/81)^(x-6)=3
- Express {x} in terms of y where:
- 5*x+5*y=20
- 2*x+13*y=19
- 14*x-16*y=14
- 15*x+15*y=16
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Identical expressions
- (one / eighty-one)^(x- six)= three
- (1 divide by 81) to the power of (x minus 6) equally 3
- (one divide by eighty minus one) to the power of (x minus six) equally three
- (1/81)(x-6)=3
- 1/81x-6=3
- 1/81^x-6=3
- (1 divide by 81)^(x-6)=3
-
Similar expressions
- (1/81)^(x+6)=3
(1/81)^(x-6)=3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$\left(\frac{1}{81}\right)^{x - 6} = 3$$
or
$$\left(\frac{1}{81}\right)^{x - 6} - 3 = 0$$
or
$$282429536481 \cdot 81^{- x} = 3$$
or
$$\left(\frac{1}{81}\right)^{x} = \frac{1}{94143178827}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{81}\right)^{x}$$
we get
$$v - \frac{1}{94143178827} = 0$$
or
$$v - \frac{1}{94143178827} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{1}{94143178827}$$
We get the answer: v = 1/94143178827
do backward replacement
$$\left(\frac{1}{81}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(81 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(\frac{1}{94143178827} \right)}}{\log{\left(\frac{1}{81} \right)}} = \frac{23}{4}$$
$$\left(\frac{1}{81}\right)^{x - 6} = 3$$
or
$$\left(\frac{1}{81}\right)^{x - 6} - 3 = 0$$
or
$$282429536481 \cdot 81^{- x} = 3$$
or
$$\left(\frac{1}{81}\right)^{x} = \frac{1}{94143178827}$$
- this is the simplest exponential equation
Do replacement
$$v = \left(\frac{1}{81}\right)^{x}$$
we get
$$v - \frac{1}{94143178827} = 0$$
or
$$v - \frac{1}{94143178827} = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = \frac{1}{94143178827}$$
We get the answer: v = 1/94143178827
do backward replacement
$$\left(\frac{1}{81}\right)^{x} = v$$
or
$$x = - \frac{\log{\left(v \right)}}{\log{\left(81 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(\frac{1}{94143178827} \right)}}{\log{\left(\frac{1}{81} \right)}} = \frac{23}{4}$$
Sum and product of roots
[src]
sum
23 log(94143178827) pi*I log(94143178827) pi*I 23 pi*I -- + ---------------- - -------- + ---------------- + -------- + -- + ------ 4 4*log(3) 2*log(3) 4*log(3) 2*log(3) 4 log(3)
$$\left(\left(\frac{23}{4} + \left(\frac{\log{\left(94143178827 \right)}}{4 \log{\left(3 \right)}} - \frac{i \pi}{2 \log{\left(3 \right)}}\right)\right) + \left(\frac{\log{\left(94143178827 \right)}}{4 \log{\left(3 \right)}} + \frac{i \pi}{2 \log{\left(3 \right)}}\right)\right) + \left(\frac{23}{4} + \frac{i \pi}{\log{\left(3 \right)}}\right)$$
=
23 log(94143178827) pi*I -- + ---------------- + ------ 2 2*log(3) log(3)
$$\frac{23}{2} + \frac{\log{\left(94143178827 \right)}}{2 \log{\left(3 \right)}} + \frac{i \pi}{\log{\left(3 \right)}}$$
product
/log(94143178827) pi*I \
23*|---------------- - --------|
\ 4*log(3) 2*log(3)/ /log(94143178827) pi*I \ /23 pi*I \
--------------------------------*|---------------- + --------|*|-- + ------|
4 \ 4*log(3) 2*log(3)/ \4 log(3)/
$$\frac{23 \left(\frac{\log{\left(94143178827 \right)}}{4 \log{\left(3 \right)}} - \frac{i \pi}{2 \log{\left(3 \right)}}\right)}{4} \left(\frac{\log{\left(94143178827 \right)}}{4 \log{\left(3 \right)}} + \frac{i \pi}{2 \log{\left(3 \right)}}\right) \left(\frac{23}{4} + \frac{i \pi}{\log{\left(3 \right)}}\right)$$
=
23*(-2*pi*I + log(94143178827))*(2*pi*I + log(94143178827))*(4*pi*I + log(94143178827))
---------------------------------------------------------------------------------------
3
256*log (3)
$$\frac{23 \left(\log{\left(94143178827 \right)} - 2 i \pi\right) \left(\log{\left(94143178827 \right)} + 2 i \pi\right) \left(\log{\left(94143178827 \right)} + 4 i \pi\right)}{256 \log{\left(3 \right)}^{3}}$$
23*(-2*pi*i + log(94143178827))*(2*pi*i + log(94143178827))*(4*pi*i + log(94143178827))/(256*log(3)^3)
Rapid solution
[src]
x1 = 23/4
$$x_{1} = \frac{23}{4}$$
log(94143178827) pi*I
x2 = ---------------- - --------
4*log(3) 2*log(3)
$$x_{2} = \frac{\log{\left(94143178827 \right)}}{4 \log{\left(3 \right)}} - \frac{i \pi}{2 \log{\left(3 \right)}}$$
log(94143178827) pi*I
x3 = ---------------- + --------
4*log(3) 2*log(3)
$$x_{3} = \frac{\log{\left(94143178827 \right)}}{4 \log{\left(3 \right)}} + \frac{i \pi}{2 \log{\left(3 \right)}}$$
23 pi*I
x4 = -- + ------
4 log(3)
$$x_{4} = \frac{23}{4} + \frac{i \pi}{\log{\left(3 \right)}}$$
x4 = 23/4 + i*pi/log(3)
Numerical answer
[src]
x1 = 5.75
x2 = 5.75 - 1.42980043369006*i
x3 = 5.75 + 1.42980043369006*i
x4 = 5.75 + 2.85960086738013*i
x4 = 5.75 + 2.85960086738013*i