What is the answer to 3^x+3^-x=6 ?

The answer to 3^x+3^-x=6 is x=-(\W(729^{3^}ln(3))-6*3^{}ln(3))/(ln(3))

Solve for x: 3^x+3^-x=6

The solution is x=-(\W(729^{3^}ln(3))-6*3^{}ln(3))/(ln(3))

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3^{x} + 3^{-} x = 6

x = - \frac{W ( 72 9 ^{3} ln ( 3 ) ) - 6 \cdot 3 ln ( 3 )}{ln ( 3 )}

Solution steps

3^{x} + 3^{-} x = 6

Prepare

3^{x} + 3^{-} x = 6

for Lambert form:

1 = (6 - 3^{-} x) e^{- l n (3) x}

Rewrite the equation with

(- x + 6 \cdot 3) ln (3) = u

and

x = - \frac{u - 6 \cdot 3 ln ( 3 )}{ln ( 3 )}

1 = (6 - 3^{-} (- \frac{u - 6 \cdot 3 ln ( 3 )}{ln ( 3 )})) e^{- l n (3) (- \frac{u - 6 \cdot 3 l n ( 3 )}{l n ( 3 )})}

Simplify

1 = e^{u - 6 \cdot 3 l n (3)} (6 + \frac{3 ^{-} ( u - 6 \cdot 3 ln ( 3 ) )}{ln ( 3 )})

Rewrite

1 = e^{u - 6 \cdot 3 l n (3)} (6 + \frac{3 ^{-} ( u - 6 \cdot 3 ln ( 3 ) )}{ln ( 3 )})

in Lambert form:

e^{u} u = 72 9^{3} ln (3)

Solve

e^{u} u = 72 9^{3} ln (3) : u = W (72 9^{3} ln (3))

u = W (72 9^{3} ln (3))

Substitute back

u = (- x + 6 \cdot 3) ln (3),

solve for

x

Solve

(- x + 6 \cdot 3) ln (3) = W (72 9^{3} ln (3)) : x = - \frac{W ( 72 9 ^{3} ln ( 3 ) ) - 6 \cdot 3 ln ( 3 )}{ln ( 3 )}

x = - \frac{W ( 72 9 ^{3} ln ( 3 ) ) - 6 \cdot 3 ln ( 3 )}{ln ( 3 )}

e^{\frac{x}{2}} = 5

6 = 5^{x}

4 e^{4 x} - 4 = 0

3 \cdot 6^{x - 1} = 2 \cdot 3^{x + 2}

What is the answer to 3^x+3^-x=6 ?
The answer to 3^x+3^-x=6 is x=-(\W(729^{3^}ln(3))-6*3^{}ln(3))/(ln(3))
Solve for x: 3^x+3^-x=6
The solution is x=-(\W(729^{3^}ln(3))-6*3^{}ln(3))/(ln(3))