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log^(2)(2x-1)=3 equation

The teacher will be very surprised to see your correct solution 😉

You have entered [src]

   2             
log (2*x - 1) = 3

$$\log{\left(2 x - 1 \right)}^{2} = 3$$

Simplify

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Sum and product of roots [src]

sum

       ___           ___
     \/ 3         -\/ 3 
1   e        1   e      
- + ------ + - + -------
2     2      2      2

$$\left(\frac{1}{2 e^{\sqrt{3}}} + \frac{1}{2}\right) + \left(\frac{1}{2} + \frac{e^{\sqrt{3}}}{2}\right)$$

       ___       ___
     \/ 3     -\/ 3 
    e        e      
1 + ------ + -------
      2         2

$$\frac{1}{2 e^{\sqrt{3}}} + 1 + \frac{e^{\sqrt{3}}}{2}$$

product

/       ___\ /        ___\
|     \/ 3 | |     -\/ 3 |
|1   e     | |1   e      |
|- + ------|*|- + -------|
\2     2   / \2      2   /

$$\left(\frac{1}{2} + \frac{e^{\sqrt{3}}}{2}\right) \left(\frac{1}{2 e^{\sqrt{3}}} + \frac{1}{2}\right)$$

     /  ___\
    2|\/ 3 |
cosh |-----|
     \  2  /

$$\cosh^{2}{\left(\frac{\sqrt{3}}{2} \right)}$$

cosh(sqrt(3)/2)^2

Rapid solution [src]

            ___
          \/ 3 
     1   e     
x1 = - + ------
     2     2

$$x_{1} = \frac{1}{2} + \frac{e^{\sqrt{3}}}{2}$$

             ___
          -\/ 3 
     1   e      
x2 = - + -------
     2      2

$$x_{2} = \frac{1}{2 e^{\sqrt{3}}} + \frac{1}{2}$$

x2 = exp(-sqrt(3))/2 + 1/2

Numerical answer [src]

x1 = 3.32611683701705

x2 = 0.588460603158882

x2 = 0.588460603158882