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3^x=-1

3^x=-1 equation

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Numerical solution:

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The solution

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 x     
3  = -1
3x=13^{x} = -1
Detail solution
Given the equation:
3x=13^{x} = -1
or
3x+1=03^{x} + 1 = 0
or
3x=13^{x} = -1
or
3x=13^{x} = -1
- this is the simplest exponential equation
Do replacement
v=3xv = 3^{x}
we get
v+1=0v + 1 = 0
or
v+1=0v + 1 = 0
Move free summands (without v)
from left part to right part, we given:
v=1v = -1
We get the answer: v = -1
do backward replacement
3x=v3^{x} = v
or
x=log(v)log(3)x = \frac{\log{\left(v \right)}}{\log{\left(3 \right)}}
The final answer
x1=log(1)log(3)=iπlog(3)x_{1} = \frac{\log{\left(-1 \right)}}{\log{\left(3 \right)}} = \frac{i \pi}{\log{\left(3 \right)}}
The graph
024-14-12-10-8-6-4-2-5050
Rapid solution [src]
      pi*I 
x1 = ------
     log(3)
x1=iπlog(3)x_{1} = \frac{i \pi}{\log{\left(3 \right)}}
x1 = i*pi/log(3)
Sum and product of roots [src]
sum
 pi*I 
------
log(3)
iπlog(3)\frac{i \pi}{\log{\left(3 \right)}}
=
 pi*I 
------
log(3)
iπlog(3)\frac{i \pi}{\log{\left(3 \right)}}
product
 pi*I 
------
log(3)
iπlog(3)\frac{i \pi}{\log{\left(3 \right)}}
=
 pi*I 
------
log(3)
iπlog(3)\frac{i \pi}{\log{\left(3 \right)}}
pi*i/log(3)
Numerical answer [src]
x1 = 2.85960086738013*i
x1 = 2.85960086738013*i
The graph
3^x=-1 equation