Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 3*x=28-x
-
Equation 3(x-7)-6x=-x
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Equation cos(x)^(2)=(1/2)
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Equation (x+3)^2-16=(1-2x)^2
- Express {x} in terms of y where:
- -9*x-19*y=13
- -13*x-5*y=6
- 14*x+3*y=-4
- -11*x+13*y=10
- Sum of series:
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1/2
- Derivative of:
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1/2
- Integral of d{x}:
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1/2
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Identical expressions
- sixteen ^(x- nine)= one / two
- 16 to the power of (x minus 9) equally 1 divide by 2
- sixteen to the power of (x minus nine) equally one divide by two
- 16(x-9)=1/2
- 16x-9=1/2
- 16^x-9=1/2
- 16^(x-9)=1 divide by 2
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Similar expressions
- 16^(x+9)=1/2
16^(x-9)=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$16^{x - 9} = \frac{1}{2}$$
or
$$16^{x - 9} - \frac{1}{2} = 0$$
or
$$\frac{16^{x}}{68719476736} = \frac{1}{2}$$
or
$$16^{x} = 34359738368$$
- this is the simplest exponential equation
Do replacement
$$v = 16^{x}$$
we get
$$v - 34359738368 = 0$$
or
$$v - 34359738368 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 34359738368$$
We get the answer: v = 34359738368
do backward replacement
$$16^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(16 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(34359738368 \right)}}{\log{\left(16 \right)}} = \frac{35}{4}$$
$$16^{x - 9} = \frac{1}{2}$$
or
$$16^{x - 9} - \frac{1}{2} = 0$$
or
$$\frac{16^{x}}{68719476736} = \frac{1}{2}$$
or
$$16^{x} = 34359738368$$
- this is the simplest exponential equation
Do replacement
$$v = 16^{x}$$
we get
$$v - 34359738368 = 0$$
or
$$v - 34359738368 = 0$$
Move free summands (without v)
from left part to right part, we given:
$$v = 34359738368$$
We get the answer: v = 34359738368
do backward replacement
$$16^{x} = v$$
or
$$x = \frac{\log{\left(v \right)}}{\log{\left(16 \right)}}$$
The final answer
$$x_{1} = \frac{\log{\left(34359738368 \right)}}{\log{\left(16 \right)}} = \frac{35}{4}$$
Rapid solution
[src]
x1 = 35/4
$$x_{1} = \frac{35}{4}$$
log(34359738368) pi*I
x2 = ---------------- - --------
4*log(2) 2*log(2)
$$x_{2} = \frac{\log{\left(34359738368 \right)}}{4 \log{\left(2 \right)}} - \frac{i \pi}{2 \log{\left(2 \right)}}$$
log(34359738368) pi*I
x3 = ---------------- + --------
4*log(2) 2*log(2)
$$x_{3} = \frac{\log{\left(34359738368 \right)}}{4 \log{\left(2 \right)}} + \frac{i \pi}{2 \log{\left(2 \right)}}$$
35 pi*I
x4 = -- + ------
4 log(2)
$$x_{4} = \frac{35}{4} + \frac{i \pi}{\log{\left(2 \right)}}$$
x4 = 35/4 + i*pi/log(2)
Sum and product of roots
[src]
sum
35 log(34359738368) pi*I log(34359738368) pi*I 35 pi*I -- + ---------------- - -------- + ---------------- + -------- + -- + ------ 4 4*log(2) 2*log(2) 4*log(2) 2*log(2) 4 log(2)
$$\left(\left(\frac{35}{4} + \left(\frac{\log{\left(34359738368 \right)}}{4 \log{\left(2 \right)}} - \frac{i \pi}{2 \log{\left(2 \right)}}\right)\right) + \left(\frac{\log{\left(34359738368 \right)}}{4 \log{\left(2 \right)}} + \frac{i \pi}{2 \log{\left(2 \right)}}\right)\right) + \left(\frac{35}{4} + \frac{i \pi}{\log{\left(2 \right)}}\right)$$
=
35 log(34359738368) pi*I -- + ---------------- + ------ 2 2*log(2) log(2)
$$\frac{35}{2} + \frac{\log{\left(34359738368 \right)}}{2 \log{\left(2 \right)}} + \frac{i \pi}{\log{\left(2 \right)}}$$
product
/log(34359738368) pi*I \
35*|---------------- - --------|
\ 4*log(2) 2*log(2)/ /log(34359738368) pi*I \ /35 pi*I \
--------------------------------*|---------------- + --------|*|-- + ------|
4 \ 4*log(2) 2*log(2)/ \4 log(2)/
$$\frac{35 \left(\frac{\log{\left(34359738368 \right)}}{4 \log{\left(2 \right)}} - \frac{i \pi}{2 \log{\left(2 \right)}}\right)}{4} \left(\frac{\log{\left(34359738368 \right)}}{4 \log{\left(2 \right)}} + \frac{i \pi}{2 \log{\left(2 \right)}}\right) \left(\frac{35}{4} + \frac{i \pi}{\log{\left(2 \right)}}\right)$$
=
35*(-2*pi*I + log(34359738368))*(2*pi*I + log(34359738368))*(4*pi*I + log(34359738368))
---------------------------------------------------------------------------------------
3
256*log (2)
$$\frac{35 \left(\log{\left(34359738368 \right)} - 2 i \pi\right) \left(\log{\left(34359738368 \right)} + 2 i \pi\right) \left(\log{\left(34359738368 \right)} + 4 i \pi\right)}{256 \log{\left(2 \right)}^{3}}$$
35*(-2*pi*i + log(34359738368))*(2*pi*i + log(34359738368))*(4*pi*i + log(34359738368))/(256*log(2)^3)
Numerical answer
[src]
x1 = 8.75
x2 = 8.75 - 2.2661800709136*i
x3 = 8.75 + 2.2661800709136*i
x4 = 8.75 + 4.53236014182719*i
x4 = 8.75 + 4.53236014182719*i
The graph